Robust Federated Learning

Robust Federated Learning (Byzantine-Tolerant Distributed Training)

Byzantine-robust federated learning
fault-tolerant federated learning
robust FL
Byzantine-tolerant distributed learning

Blanchard, P.; El Mhamdi, E. M.; Guerraoui, R.

Robust Federated Learning extends standard federated learning with Byzantine-tolerant aggregation rules that protect the global model against malicious, corrupted, or unreliable clients. Instead of naively averaging client gradients, robust aggregation methods such as coordinate-wise median or Krum filter out harmful updates so that a minority of adversarial participants cannot derail training.

Distributed learning with Byzantine-tolerant aggregation

Decentralized / partitioned tabular, image, or text data across clients

Blanchard, P., El Mhamdi, E. M., Guerraoui, R., & Stainer, J. (2017). Machine Learning with Adversaries: Byzantine Tolerant Gradient Descent. Advances in Neural Information Processing Systems, 30.

Yin, D., Chen, Y., Kannan, R., & Bartlett, P. (2018). Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates. Proceedings of the 35th International Conference on Machine Learning (ICML), PMLR 80:5650–5659.

federated-learning semi-supervised-federated-learning online-federated-learning bayesian-federated-learning transfer-learning robust-gradient-boosting


Robust Fisher's exact test

robust exact test for contingency tables

1935 (base); mid-p robustification 1961+

Fisher (1935); mid-p extension by Lancaster (1961) and others

The robust Fisher's exact test extends Fisher's classic exact test for contingency tables by applying conservative-correcting adjustments — most commonly the mid-p correction — to reduce the extreme conservatism of the standard exact test. This produces better-calibrated Type I error rates while maintaining validity in small and sparse samples.

Categorical (2×2 or small r×c contingency tables)

Agresti, A. (2002). Categorical Data Analysis (2nd ed.). Wiley-Interscience.

Lancaster, H. O. (1961). Significance tests in discrete distributions. Journal of the American Statistical Association, 56(294), 223–234.

fishers-exact-test chi-square-test robust-chi-square-test barnard-exact-test mid-p-correction bootstrap-fishers-exact-test


Robust Fixed Effects Model

fixed effects with heteroscedasticity-robust SE

The robust fixed effects model combines the within-group estimator for panel data with variance-covariance matrices that remain valid under heteroscedasticity and within-unit error correlation. Introduced by Arellano (1987), cluster-robust standard errors paired with the fixed effects estimator are now the default approach for credible panel data inference in economics and social science.

Balanced or unbalanced panel data (cross-sectional units observed over time)

Arellano, M. (1987). Computing robust standard errors for within-groups estimators. Oxford Bulletin of Economics and Statistics, 49(4), 431–434.

Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press.

fixed-effects-model random-effects-model panel-hausman-test robust-ols panel-ols panel-random-effects-model


Robust Fractional Factorial Design

Robust Parameter Design with Fractional Factorial Arrays

robust FFD
robust fractional factorial experiment
crossed-array fractional factorial
Taguchi-style fractional factorial

1980s (Taguchi's crossed-array approach); fractional factorial roots 1935–1945

Genichi Taguchi (robust parameter design); fractional factorial foundations by Ronald Fisher and Frank Yates

Robust fractional factorial design combines the run-count efficiency of fractional factorial arrays with Taguchi's robust parameter design philosophy. By simultaneously manipulating control factors (inner array) and noise factors (outer array) — each structured as a fractional factorial — the method identifies factor settings that minimize product or process variation due to uncontrollable conditions, without requiring a full factorial experiment.

Experimental design / robust parameter design

Continuous or ordinal response measurements from controlled experiments

Montgomery, D. C. (2017). Design and Analysis of Experiments (9th ed.). Wiley.

Taguchi, G. (1987). System of Experimental Design: Engineering Methods to Optimize Quality and Minimize Costs. UNIPUB/Kraus International.

fractional-factorial-design taguchi-method full-factorial-design response-surface-methodology robust-design-of-experiments central-composite-design


Robust frequency analysis

robust count analysis
outlier-resistant frequency analysis
robust distributional analysis

1970s–1980s (foundations); applied to frequency analysis throughout the 1990s–2000s

Huber, Hampel, Wilcox and the robust statistics tradition

Robust frequency analysis applies outlier-resistant estimation and resampling or exact methods to the counting and tabulation of categorical data, reducing the distortion caused by extreme observations, sparse cells, or violations of large-sample assumptions that can make conventional frequency summaries misleading.

Robust descriptive and inferential procedure

Categorical or ordinal counts; frequency tables

Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

Huber, P. J., & Ronchetti, E. M. (2009). Robust Statistics (2nd ed.). Wiley.

frequency-analysis robust-chi-square-test robust-descriptive-statistics bootstrap-frequency-analysis permutation-frequency-analysis robust-cross-tabulation-analysis


Robust Friedman test

Robust Friedman Test for Repeated Measures

robust rank-based repeated measures test

Extension of Friedman (1937); robust variants developed by Wilcox and colleagues

The robust Friedman test is a nonparametric procedure for comparing three or more related (within-subjects) conditions that replaces standard ranking or mean-based summaries with robust location estimates — typically trimmed means or Winsorized statistics — to reduce the influence of outliers and heavy-tailed distributions on the inference.

Robust nonparametric repeated measures comparison

Continuous or ordinal, repeated measures (within-subjects), three or more conditions

Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association, 32(200), 675–701.

friedman-test robust-repeated-measures-anova robust-wilcoxon-signed-rank-test robust-kruskal-wallis-test repeated-measures-anova robust-paired-samples-t-test


Robust Full Factorial Design

Robust Full Factorial Design of Experiments

robust 2^k design
full factorial robust parameter design
robust FFD
noise-factor full factorial

Genichi Taguchi (robustness principles); formalized in combined-array form by Shoemaker, Tsui, and Wu (1991)

Robust full factorial design extends the classical full factorial experiment by explicitly including noise factors — uncontrollable variables that cause performance variation in real-world conditions. By crossing all control factor levels with all noise factor levels in a single combined array, engineers identify control factor settings that maximize mean performance while minimizing sensitivity to noise, yielding products and processes that perform consistently across operating environments.

Experimental design with noise-factor control

Continuous response measurements under controlled control and noise factor combinations

Phadke, M. S. (1989). Quality Engineering Using Robust Design. Prentice Hall.

full-factorial-design taguchi-method robust-taguchi-method fractional-factorial-design robust-fractional-factorial-design response-surface-methodology


Robust Fuzzy Regression Discontinuity

Robust Bias-Corrected Fuzzy Regression Discontinuity Design

Robust Fuzzy RDD
Fuzzy RD with robust inference
bias-corrected fuzzy RD
CCT fuzzy RDD

2014 (robust CCT estimator); 2001 (fuzzy RDD formalization)

Calonico, Cattaneo, and Titiunik (robust inference framework); Hahn, Todd, and Van der Klaauw (fuzzy RDD formalization)

Robust Fuzzy Regression Discontinuity Design estimates a local average treatment effect (LATE) at a threshold where crossing the cutoff raises — but does not guarantee — treatment receipt. Introduced by Calonico, Cattaneo, and Titiunik (2014), the robust framework applies bias-corrected local polynomial estimation with a robust variance estimator, correcting the coverage failures of conventional bandwidth-optimal inference in both the sharp and fuzzy cases.

Quasi-experimental causal inference with IV at threshold

Cross-sectional or panel; continuous running variable with known assignment threshold; binary or partial compliance

Calonico, S., Cattaneo, M. D., & Titiunik, R. (2014). Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs. Econometrica, 82(6), 2295-2326.

Imbens, G. W., & Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. Journal of Econometrics, 142(2), 615-635.

fuzzy-regression-discontinuity regression-discontinuity-design instrumental-variables local-average-treatment-effect propensity-score-matching difference-in-differences


Robust GARCH model

Robust Generalized Autoregressive Conditional Heteroscedasticity Model

Robust GARCH
outlier-robust GARCH
heavy-tail GARCH
contamination-robust volatility model

Boudt, Danielsson & Laurent (robust extensions); Bollerslev (standard GARCH, 1986)

The Robust GARCH model extends the classical GARCH framework to handle outliers and heavy-tailed innovations that commonly appear in financial return series. By down-weighting extreme observations through a robust innovation term, it produces more reliable volatility forecasts when data contain jumps, crises, or other anomalies that would otherwise distort standard GARCH estimates.

Boudt, K., Danielsson, J., & Laurent, S. (2013). Robust forecasting of dynamic conditional correlation GARCH models. International Journal of Forecasting, 29(2), 244–257.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.

garch-model egarch-model gjr-garch-model stochastic-volatility-model arch-model quantile-regression


Robust Gaussian Mixture Model

Robust Gaussian Mixture Model (Heavy-Tailed and Trimmed Variants)

Robust GMM
mixture of t-distributions
trimmed GMM
heavy-tailed mixture model

Robust Gaussian Mixture Model replaces the standard Gaussian components with heavier-tailed distributions — most commonly Student's t-distributions — or incorporates trimming and down-weighting of outliers within the EM framework. The result is a probabilistic clustering and density-estimation method that assigns genuinely anomalous points less influence on component parameters, preventing outliers from distorting cluster shapes or positions.

Probabilistic clustering / density estimation

Peel, D. & McLachlan, G. J. (2000). Robust mixture modelling using the t distribution. Statistics and Computing, 10(4), 339–348.

Maronna, R. A., Martin, R. D. & Yohai, V. J. (2006). Robust Statistics: Theory and Methods. Wiley.

gaussian-mixture-model k-means robust-k-means one-class-svm isolation-forest robust-linear-regression


Robust Gaussian Process

Robust Gaussian Process Regression and Classification

Robust GP
Student-t Process
Heavy-tailed Gaussian Process
Outlier-robust GP

2011 (formal treatment); GP foundations: Rasmussen & Williams 2006

Robust Gaussian Process (Robust GP) extends the standard Gaussian Process framework by replacing the Gaussian noise likelihood with a heavy-tailed distribution — typically Student-t — so that outliers in the training data exert less influence on the learned function. It retains the full probabilistic, uncertainty-quantifying character of a standard GP while becoming far less sensitive to corrupted or anomalous observations.

Probabilistic non-parametric regression / classification

Continuous, mixed; small-to-medium tabular datasets

Jylanki, P., Vanhatalo, J., & Vehtari, A. (2011). Robust Gaussian Process Regression with a Student-t Likelihood. Journal of Machine Learning Research, 12, 3227–3257.

Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press.

gaussian-process bayesian-gaussian-process robust-support-vector-machine robust-linear-regression robust-random-forest gaussian-mixture-model


Robust Geary's C

robust Geary contiguity ratio
outlier-resistant Geary's C
robust spatial contiguity statistic
robust Geary C

1954 (base); robust variants: 1990s–2000s

Geary (1954); robust extensions by Anselin and spatial statisticians

Robust Geary's C adapts the classical Geary contiguity ratio — a measure of spatial autocorrelation based on pairwise squared differences between neighbouring locations — to resist distortion by spatial outliers and influential observations. It retains the local sensitivity of Geary's C while producing more reliable inferences when the spatial data contain extreme values or non-normal distributions.

Robust spatial autocorrelation statistic

Georeferenced continuous or ordinal areal/point data

Geary, R. C. (1954). The contiguity ratio and statistical mapping. The Incorporated Statistician, 5(3), 115–145.

Anselin, L. (1995). Local indicators of spatial association — LISA. Geographical Analysis, 27(2), 93–115.


Robust Generalized linear model

robust GLM
GLM with robust estimation
robust quasi-likelihood model
M-estimator GLM

A Robust Generalized Linear Model fits the standard GLM family — linear, logistic, Poisson, and others — using M-type estimating equations that down-weight outlying or influential observations. The result is coefficient estimates and standard errors that remain stable even when a minority of data points deviate sharply from the assumed distribution.

Continuous, binary, count, or proportion outcomes with potential outliers or model contamination

Heritier, S., Cantoni, E., Copt, S., & Victoria-Feser, M.-P. (2009). Robust Methods in Biostatistics. Wiley.

Cantoni, E., & Ronchetti, E. (2001). Robust inference for generalized linear models. Journal of the American Statistical Association, 96(455), 1022–1030.

generalized-linear-model robust-regression robust-multiple-linear-regression robust-logistic-regression robust-poisson-regression quasi-likelihood-regression


Robust Genetic Algorithm

Robust Genetic Algorithm — Evolutionary Optimization under Uncertainty

RGA
Robust GA
Uncertainty-Aware Genetic Algorithm
Noise-Tolerant Genetic Algorithm

2005 (systematic survey); earlier applications from late 1990s

Jin, Y. and Branke, J. (systematic formalization); roots in Holland (1975)

The Robust Genetic Algorithm (RGA) extends standard genetic algorithms to find solutions that perform well not only at the nominal design point but also when subjected to uncertainty in decision variables, parameters, or fitness evaluations. By incorporating explicit robustness measures into selection pressure, RGA balances optimality against sensitivity to perturbation, making it suitable for engineering design, scheduling, and policy optimization under real-world variability.

Metaheuristic evolutionary optimizer with robustness mechanism

Continuous or discrete decision variables with uncertain objective evaluations

Jin, Y., Branke, J. (2005). Evolutionary optimization in uncertain environments — a survey. IEEE Transactions on Evolutionary Computation, 9(3), 303–317.

Beyer, H.-G., Sendhoff, B. (2007). Robust optimization — A comprehensive survey. Computer Methods in Applied Mechanics and Engineering, 196(33–34), 3190–3218.

genetic-algorithm multi-objective-genetic-algorithm robust-multi-objective-optimization stochastic-genetic-algorithm robust-particle-swarm-optimization robust-simulated-annealing


Robust Getis-Ord Gi*

robust local spatial autocorrelation Gi*

1992 (base); robust variants circa 2000s–2010s

Getis & Ord (base statistic); robust extensions developed in subsequent spatial statistics literature

The Robust Getis-Ord Gi* statistic extends the classical Gi* hot-spot measure to handle outliers in spatial data. By using robust estimators of the mean and variance — such as trimmed means, medians, or down-weighted influential observations — it identifies statistically significant spatial clusters of high or low values even when the attribute distribution contains extreme values that would distort the standard Gi*.

Georeferenced areal or point data with a continuous attribute

Getis, A., & Ord, J. K. (1992). The analysis of spatial association by use of distance statistics. Geographical Analysis, 24(3), 189–206.

Anselin, L., & Liu, X. (2010). Spatial panel econometrics. In Handbook of Applied Economic Statistics. Robust spatial statistics variants are discussed in the context of outlier-resistant local indicators. See also: Anselin, L. (2018). A local indicator of multivariate spatial association. Geographical Analysis, 51(2), 133–150.


Robust Gibbs Sampling

robust MCMC Gibbs sampler
outlier-resistant Gibbs sampling
heavy-tailed Gibbs sampler
robust block Gibbs

Stuart Geman & Donald Geman (Gibbs sampler, 1984); robustness extensions developed through 1990s Bayesian literature

Robust Gibbs sampling is a Markov chain Monte Carlo strategy that pairs the coordinate-wise Gibbs sampler with heavy-tailed or outlier-resistant model specifications — most commonly Student-t likelihoods — so that the posterior inference is not distorted by extreme observations. It achieves robustness through data augmentation: each observation receives a latent variance weight that automatically down-weights outliers during each sampling sweep.

Continuous, count, or mixed data with potential outliers or heavy-tailed errors

Geweke, J. (1993). Bayesian treatment of the independent Student-t linear model. Journal of Applied Econometrics, 8(S1), S19–S40.

Chib, S. & Greenberg, E. (1995). Understanding the Metropolis-Hastings algorithm. The American Statistician, 49(4), 327–335.

gibbs-sampling robust-bayesian-inference robust-markov-chain-monte-carlo hierarchical-gibbs-sampling robust-metropolis-hastings-algorithm bayesian-regression


Robust GLS

robust generalized least squares
GLS with robust standard errors
heteroscedasticity-consistent GLS
HC-GLS

Aitken (GLS theory, 1936); White (robust covariance, 1980)

Robust GLS extends classical Generalized Least Squares by pairing GLS coefficient estimation with heteroscedasticity- and autocorrelation-consistent (HAC) standard errors, or by using M-estimation within the GLS framework. It corrects for non-spherical errors — heteroscedasticity, autocorrelation, or both — while also guarding inference against misspecification of the error covariance structure.

Cross-sectional or time-series continuous data with non-spherical errors

Greene, W. H. (2012). Econometric Analysis (7th ed.). Pearson. Chapter 9: The Generalized Regression Model and Heteroscedasticity.

White, H. (1980). A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity. Econometrica, 48(4), 817-838.

ols-regression panel-gls robust-ols feasible-gls weighted-least-squares generalized-least-squares


Robust goal programming

RGP
Goal Programming under Uncertainty
Robust GP
Uncertainty-Aware Goal Programming

Charnes, A. & Cooper, W. W. (goal programming); Mulvey, J. M. et al. (robust optimization framework)

Robust Goal Programming (RGP) extends classical goal programming to handle uncertain or ambiguous model parameters. Instead of minimizing deviations from crisp targets, it seeks solutions that remain feasible and near-optimal across a range of plausible scenarios or uncertain data realizations. RGP is particularly valuable in planning problems where goals are aspirational and input data carries inherent variability or estimation error.

Mathematical programming under uncertainty

Numerical targets, uncertain coefficients (interval or scenario-based)

Charnes, A., Cooper, W. W. (1961). Management Models and Industrial Applications of Linear Programming. Wiley, New York.

Mulvey, J. M., Vanderbei, R. J., Zenios, S. A. (1995). Robust optimization of large-scale systems. Operations Research, 43(2), 264-281.

goal-programming stochastic-goal-programming robust-linear-programming robust-multi-objective-optimization multi-objective-goal-programming stochastic-programming


Robust Gradient Boosting

Robust Gradient Boosting (Gradient Boosting with Robust Loss Functions)

gradient boosting with Huber loss
robust GBM
outlier-robust boosting
robust gradient-boosted trees

Friedman, J. H. (with Huber loss from Huber, P. J.)

Robust Gradient Boosting is gradient boosting trained with outlier-resistant loss functions — most commonly the Huber loss or quantile (pinball) loss — instead of squared-error loss. Proposed in Friedman's seminal 2001 paper, this variant produces predictions far less distorted by extreme values or contaminated labels, while retaining the full predictive power of gradient-boosted trees.

Ensemble (boosted trees with robust loss)

Tabular, continuous/categorical features; continuous or ordinal target

Friedman, J. H. (2001). Greedy function approximation: A gradient boosting machine. Annals of Statistics, 29(5), 1189–1232.

Huber, P. J. (1964). Robust Estimation of a Location Parameter. Annals of Mathematical Statistics, 35(1), 73–101.

boosting gradient-boosting xgboost robust-linear-regression regularized-gradient-boosting random-forest


Robust Granger Causality

2006 (robust variant); 1969 (original Granger)

Hacker & Hatemi-J (robust bootstrap variant); Granger (original causality concept)

Robust Granger causality extends the classic Granger causality framework by using bootstrap-based or heteroscedasticity-robust critical values rather than asymptotic chi-squared tables. This makes the test reliable in finite samples and when the data exhibit non-normality, heteroscedasticity, or near-integration, settings where the standard F- or Wald-based test is known to over-reject.

Time series (possibly integrated or heteroscedastic)

Hacker, R. S., & Hatemi-J, A. (2006). Tests for causality between integrated variables using asymptotic and bootstrap distributions: Theory and application. Applied Economics, 38(13), 1489–1500.

Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37(3), 424–438.

granger-causality toda-yamamoto-causality var-model cointegration-test bootstrap-causality vecm


Robust Hamiltonian Monte Carlo

Robust HMC
heavy-tailed HMC
geometric-ergodic HMC
outlier-robust HMC

Livingstone, Zanella and related researchers building on Duane et al. (1987)

Robust Hamiltonian Monte Carlo (Robust HMC) is a family of extensions to standard HMC designed to maintain geometric ergodicity and sampling efficiency when the posterior has heavy tails, strong curvature variation, or near-degenerate geometry. By modifying the kinetic energy, mass matrix, or proposal mechanism, these methods ensure reliable exploration of difficult posteriors that defeat the standard NUTS/HMC sampler.

Continuous parameters; posteriors with heavy tails or challenging geometry

Livingstone, S. & Zanella, G. (2022). The Barker proposal: combining robustness and efficiency in gradient-based MCMC. Journal of the Royal Statistical Society: Series B, 84(2), 496–523.

Betancourt, M. (2017). A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434.

hamiltonian-monte-carlo metropolis-hastings nuts-sampler robust-bayesian-inference gibbs-sampling variational-inference


Robust Hausman Test

Heteroscedasticity- and Autocorrelation-Robust Hausman Specification Test

robust hausman specification test
cluster-robust hausman test
Robust Hausman Testi

Hausman (1978); robust variant after Arellano (1993)

The Robust Hausman Test is a heteroscedasticity- and autocorrelation-robust version of the Hausman specification test, used to choose between fixed-effects and random-effects estimators in panel-data models. It builds on Hausman's 1978 test and the robust treatment of correlated effects developed by Arellano (1993).

Robust Wald-type contrast of fixed- vs random-effects estimates

Hausman, J. A. (1978). Specification Tests in Econometrics. Econometrica, 46(6), 1251-1271.

Arellano, M. (1993). On the Testing of Correlated Effects with Panel Data. Journal of Econometrics, 59(1-2), 87-97.

panel-fixed-effects panel-random-effects ols-regression wild-bootstrap permutation-test


Robust HDBSCAN

Robust Hierarchical Density-Based Spatial Clustering of Applications with Noise

HDBSCAN*
Robust HDBSCAN*
robust hierarchical density clustering
robust single-linkage HDBSCAN

Campello, R.J.G.B.; Moulavi, D.; Zimek, A.; Sander, J.

Robust HDBSCAN (HDBSCAN*) extends the original HDBSCAN algorithm with a robust single-linkage framework that handles noise, outliers, and clusters of varying densities more reliably. Introduced by Campello et al. (2015), it converts any density-based hierarchy into a stable flat clustering while explicitly modeling noise points — without requiring the user to pre-specify the number of clusters.

Hierarchical density-based clustering with robust single-linkage

Continuous, high-dimensional, or noisy tabular data

Campello, R.J.G.B., Moulavi, D., Zimek, A. & Sander, J. (2015). Hierarchical Density Estimates for Data Clustering, Visualization, and Outlier Detection. ACM Transactions on Knowledge Discovery from Data, 10(1), 5.

McInnes, L., Healy, J. & Astels, S. (2017). hdbscan: Hierarchical density based clustering. Journal of Open Source Software, 2(11), 205.

hdbscan dbscan optics-clustering k-means gaussian-mixture-model spectral-clustering


Robust Hierarchical Clustering

Kaufman & Rousseeuw (building on Ward, 1963 and others)

Robust hierarchical clustering extends classical agglomerative or divisive hierarchical clustering by replacing sensitive distance measures and linkage criteria with outlier-resistant alternatives, preserving cluster structure even when data contain anomalous observations or heavy-tailed distributions.

Continuous or mixed multivariate data, possibly containing outliers

Kaufman, L. & Rousseeuw, P. J. (1990). Finding Groups in Data: An Introduction to Cluster Analysis. Wiley.

Garcia-Escudero, L. A., Gordaliza, A., Matran, C. & Mayo-Iscar, A. (2010). A review of robust clustering methods. Advances in Data Analysis and Classification, 4(2–3), 89–109.

cluster-analysis robust-k-means-clustering hierarchical-clustering multidimensional-scaling robust-principal-component-analysis mixture-modeling


Robust Hierarchical Linear Model

robust HLM
robust multilevel model
robust mixed-effects linear model
robust nested regression

Maas & Hox (2004); Goldstein et al. (2018)

Robust Hierarchical Linear Model (Robust HLM) extends standard HLM by replacing or protecting its standard errors against violations of distributional assumptions — chiefly non-normal residuals, heteroscedasticity, and influential clusters. It retains the nested, two-level (or higher) structure while producing more trustworthy inference under real-world data conditions.

Continuous outcome, clustered / nested observations

Maas, C. J. M., & Hox, J. J. (2004). Robustness issues in multilevel regression analysis. Statistica Neerlandica, 58(2), 127–137.

Hox, J. J. (2010). Multilevel Analysis: Techniques and Applications (2nd ed.). Routledge.

hierarchical-linear-model mixed-effects-model multilevel-modeling robust-mixed-effects-model robust-multiple-linear-regression robust-regression


Robust independent samples t-test

Robust Independent Samples t-test (Trimmed Means / Winsorized Variances)

Yuen's t-test
trimmed-mean t-test
Winsorized t-test
robust two-sample test

Rand R. Wilcox; Karen K. Yuen (trimmed-mean form)

The robust independent samples t-test compares the central tendency of two independent groups using trimmed means and Winsorized variances, making it substantially less sensitive to outliers and non-normality than the classical Student or Welch t-test. The most widely used form is Yuen's test, which also accommodates unequal variances across groups.

Continuous (two independent groups, non-normal or outlier-prone distributions)

Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61(1), 165–170.


Robust Instrumental Variables

Robust Instrumental Variables Estimation

Robust IV
Weak-instrument-robust IV
Robust 2SLS
Weak-instrument-robust inference

Anderson & Rubin (1949); Stock, Wright & Yogo (2002); Andrews, Stock & Sun (2019)

Robust Instrumental Variables estimation extends standard IV and two-stage least squares (2SLS) by guarding against weak-instrument bias and non-standard inference. Methods such as the Anderson-Rubin test, Limited Information Maximum Likelihood (LIML), and the Conditional Likelihood Ratio test provide valid confidence sets and hypothesis tests even when instruments are weak or only partially identified, making IV inference reliable in settings where standard 2SLS breaks down.

Cross-sectional, panel, or time-series data with endogenous regressors and instruments

Stock, J. H., Wright, J. H., & Yogo, M. (2002). A survey of weak instruments and weak identification in generalized method of moments. Journal of Business and Economic Statistics, 20(4), 518-529.

Andrews, I., Stock, J. H., & Sun, L. (2019). Weak instruments in instrumental variables regression: Theory and practice. Annual Review of Economics, 11, 727-753.

instrumental-variables two-stage-least-squares limited-information-maximum-likelihood difference-in-differences regression-discontinuity-design panel-data-instrumental-variables


Robust Integer Programming

Robust Integer Programming — Optimization under uncertainty with integrality constraints

RIP
Robust IP
Robust Combinatorial Optimization
Integer Robust Optimization

Robust Integer Programming (RIP) finds integer or binary solutions that remain feasible and near-optimal across all scenarios in a prescribed uncertainty set. Rather than assuming exact knowledge of data, RIP hedges against the worst-case realization of uncertain costs or constraint coefficients, delivering decisions that are guaranteed to perform well even when inputs deviate from their nominal values.

Deterministic robust optimization with integer variables

Deterministic cost/constraint coefficients with uncertainty sets; integer or binary decision variables

Bertsimas, D., Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Programming, 98(1-3), 49-71.

Ben-Tal, A., El Ghaoui, L., Nemirovski, A. (2009). Robust Optimization. Princeton University Press, Princeton, NJ.

integer-programming robust-linear-programming stochastic-integer-programming mixed-integer-programming robust-mixed-integer-programming robust-multi-objective-optimization


Robust Interrupted Time Series

robust ITS
outlier-robust ITS
robust segmented regression
robust ITSA

Bernal, Cummins & Gasparrini; Linden (robust extensions)

Robust Interrupted Time Series Analysis is a quasi-experimental method that estimates the causal effect of a policy or intervention on an aggregate outcome over time, using segmented regression fitted with outlier-resistant or heteroskedasticity-consistent standard errors. It is widely used in health services research and public-health evaluation when the time series contains influential observations, non-constant variance, or mild autocorrelation.

Quasi-experimental segmented regression with robust inference

Longitudinal / time-series (aggregate or panel)

Bernal, J. L., Cummins, S., & Gasparrini, A. (2017). Interrupted time series regression for the evaluation of public health interventions: a tutorial. International Journal of Epidemiology, 46(1), 348-355.

Linden, A. (2015). Conducting interrupted time-series analysis for single- and multiple-group comparisons. Stata Journal, 15(2), 480-500.

interrupted-time-series difference-in-differences segmented-regression panel-data-interrupted-time-series robust-difference-in-differences dynamic-interrupted-time-series


Robust Inverse Probability Weighting

Robust Inverse Probability Weighting Estimator

Robust IPW
Stabilized IPW
Trimmed IPW
Variance-robust IPW

Lunceford & Davidian (2004); Robins, Hernán & Brumback (2000)

Robust Inverse Probability Weighting is a causal inference estimator that reweights observed units by stabilized or trimmed propensity score weights, then applies sandwich or bootstrap variance estimation to guard against model misspecification, extreme weights, and inflated standard errors. It extends standard IPW to improve finite-sample performance and inferential reliability in observational studies.

Observational cross-sectional or panel data with binary treatment

Lunceford, J. K., & Davidian, M. (2004). Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. Statistics in Medicine, 23(19), 2937-2960.

Robins, J. M., Hernán, M. A., & Brumback, B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology, 11(5), 550-560.

inverse-probability-weighting propensity-score-weighting doubly-robust-estimation marginal-structural-model propensity-score-matching augmented-inverse-probability-weighting


Robust Isolation forest

Robust Isolation Forest (Anomaly Detection with Robustness to Noise and Contamination)

Robust iForest
noise-robust isolation forest
contamination-robust isolation forest
robust anomaly isolation

Liu, F. T., Ting, K. M., Zhou, Z.-H. (base); robust extensions by multiple authors

Robust Isolation Forest extends the classic Isolation Forest anomaly detector with strategies that reduce sensitivity to data contamination, masking effects, and biased random splits. By incorporating robustness mechanisms — such as improved subsampling, re-weighting of suspicious regions, or bias-corrected splitting — it achieves more reliable anomaly scores when the training data itself contains a non-trivial fraction of anomalies or when specific feature distributions cause standard iForest to produce unreliable path lengths.

Liu, F. T., Ting, K. M., & Zhou, Z.-H. (2008). Isolation Forest. In Proceedings of the IEEE International Conference on Data Mining (ICDM), pp. 413–422. IEEE.

Hariri, S., Kind, M. C., & Brunner, R. J. (2019). Extended Isolation Forest. IEEE Transactions on Knowledge and Data Engineering, 33(4), 1479–1489.

isolation-forest one-class-svm autoencoder-anomaly-detection gaussian-mixture-model robust-one-class-svm robust-autoencoder-anomaly-detection


Robust Item Analysis

robust item statistics
outlier-resistant item analysis
robust classical item analysis

Robust methods tradition (Huber, Hampel, Tukey); applied to item analysis by Wilcox and colleagues

Robust item analysis applies outlier-resistant statistical methods to the evaluation of individual test or scale items. Instead of classical means and Pearson correlations — both sensitive to extreme scores — it uses trimmed means, Winsorized correlations, or M-estimators to obtain item difficulty and item-total discrimination indices that remain stable when respondent distributions are skewed or contaminated by outliers.

Ordinal or continuous item scores; polytomous or dichotomous responses

Huber, P. J. & Ronchetti, E. M. (2009). Robust Statistics (2nd ed.). Wiley.

item-response-theory cronbachs-alpha scale-development differential-item-functioning robust-reliability-analysis exploratory-factor-analysis


Robust Johansen Cointegration

outlier-robust Johansen test
robust trace test
robust maximum eigenvalue test
robust cointegration rank test

Johansen (1988, 1991); robust extensions by Cavaliere, Rahbek, Taylor (2010) and others

The Robust Johansen Cointegration test extends the classical Johansen (1988, 1991) likelihood-ratio framework for determining the cointegrating rank of a multivariate I(1) system to settings where standard Gaussian assumptions fail — in particular when the data exhibit outliers, fat-tailed innovations, or conditional heteroskedasticity. Robust modifications adjust residuals, re-weight observations, or bootstrap critical values so that rank inference remains valid under these violations.

Cointegration rank test (robust variant)

Multivariate I(1) time series, possibly with outliers or conditional heteroskedasticity

Johansen, S. (1991). Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. Econometrica, 59(6), 1551–1580.

Cavaliere, G., Rahbek, A., & Taylor, A. M. R. (2010). Cointegration Rank Testing under Conditional Heteroskedasticity. Econometric Theory, 26(6), 1719–1760.

johansen-cointegration-test engle-granger-cointegration-test vector-error-correction-model robust-engle-granger-cointegration structural-break-johansen-cointegration panel-johansen-cointegration


Robust K-means Clustering

trimmed k-means
TCLUST k-means
contamination-resistant k-means
outlier-robust clustering

Robust K-means clustering is an extension of classical k-means that protects cluster estimates from distortion caused by outliers or contaminated observations. By trimming a user-specified fraction of the most extreme points before updating cluster centers, the algorithm yields stable, meaningful partitions even when the data contain atypical cases that would severely bias standard k-means.

Continuous multivariate data with potential outliers or contamination

Cuesta-Albertos, J. A., Gordaliza, A., & Matrán, C. (1997). Trimmed k-means: An attempt to robustify quantizers. The Annals of Statistics, 25(2), 553–576.

García-Escudero, L. A., Gordaliza, A., Matrán, C., & Mayo-Iscar, A. (2008). A general trimming approach to robust cluster analysis. The Annals of Statistics, 36(3), 1324–1345.

cluster-analysis robust-hierarchical-clustering mixture-modeling robust-mixture-modeling principal-component-analysis robust-principal-component-analysis


Robust k-means

robust k-means clustering
trimmed k-means
outlier-resistant k-means
RKM

Robust k-means is a variant of classical k-means clustering designed to resist the influence of outliers. By trimming a specified fraction of the most extreme observations before computing cluster centers, it produces stable and meaningful partitions even when the data contain noise, contamination, or heavy-tailed distributions — situations where standard k-means breaks down.

Garcia-Escudero, L. A., & Gordaliza, A. (1999). Robustness properties of k-means and trimmed k-means. Journal of the American Statistical Association, 94(447), 956–969.

Garcia-Escudero, L. A., Gordaliza, A., Matrán, C., & Mayo-Iscar, A. (2008). A general trimming approach to robust cluster analysis. Annals of Statistics, 36(3), 1324–1345.

k-means k-medoids dbscan gaussian-mixture-model hierarchical-clustering spectral-clustering


Robust Kalman Filter

RKF
heavy-tailed Kalman filter
outlier-robust Kalman filter
robust state estimation

Derived from Kalman (1960); robust extensions developed by Masreliez, Martin, and others from the 1970s onward

The Robust Kalman Filter is an extension of the classical Kalman filter designed to maintain reliable state estimation when observations or process noise depart from the Gaussian assumption — particularly when data contain outliers, heavy-tailed distributions, or gross errors. By replacing or downweighting the standard least-squares update with influence-limited or M-estimation-based corrections, it prevents a single anomalous measurement from distorting the entire state estimate.

Sequential Bayesian state estimator with robustified update step

Time-series state-space observations, potentially containing outliers or heavy-tailed noise

Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1), 35-45.

Huber, P. J. & Ronchetti, E. M. (2011). Robust Statistics (2nd ed.). Wiley.

kalman-filter particle-filter robust-bayesian-inference sequential-monte-carlo robust-metropolis-hastings-algorithm extended-kalman-filter


Robust Kendall's tau

robust tau
skipped Kendall's tau
Winsorized Kendall's tau
outlier-resistant rank correlation

Rand Wilcox; Croux & Dehon (robust extensions)

Robust Kendall's tau estimates the monotone association between two variables using rank-based concordance counts, but augments the standard procedure with outlier detection or Winsorization so that a small number of extreme observations cannot distort the result. It is appropriate when data are ordinal or continuous and bivariate outliers are plausible.

Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

Croux, C., & Dehon, C. (2010). Influence functions of the Spearman and Kendall correlation measures. Statistical Methods & Applications, 19(4), 497–515.

kendalls-tau robust-spearman-correlation robust-pearson-correlation spearman-correlation mann-whitney-u-test robust-mann-whitney-u-test


Robust KPSS test

Robust KPSS
outlier-robust stationarity test
robust LM stationarity test
KPSS with robustness correction

Extension building on Kwiatkowski, Phillips, Schmidt & Shin (1992); robust variants developed by Hobijn, Franses & Ooms and others

The Robust KPSS test is an extension of the classical Kwiatkowski-Phillips-Schmidt-Shin (1992) stationarity test that replaces the conventional long-run variance estimator with an outlier-robust or heteroscedasticity-robust counterpart, maintaining reliable size and power in the presence of contaminated observations, structural breaks, or non-standard error distributions.

Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54(1-3), 159-178.

Hobijn, B., Franses, P. H., & Ooms, M. (2004). Generalizations of the KPSS-test for stationarity. Statistica Neerlandica, 58(4), 483-502.

kpss-test adf-test pp-test za-unit-root-test ls-unit-root-test df-gls-test


Robust Kriging

robust spatial kriging
outlier-resistant kriging
resistant kriging
robust geostatistical interpolation

Robust Kriging is a geostatistical interpolation method that extends classical kriging by replacing sensitive variogram estimation with outlier-resistant alternatives, most notably the Cressie-Hawkins robust estimator. It produces spatially interpolated predictions that are not distorted by anomalous or extreme observations in the data.

Continuous georeferenced point data with potential outliers

Cressie, N., & Hawkins, D. M. (1980). Robust estimation of the variogram: I. Journal of the International Association for Mathematical Geology, 12(2), 115–125.

Cressie, N. A. C. (1993). Statistics for Spatial Data (Revised ed.). Wiley-Interscience.

ordinary-kriging universal-kriging co-kriging kernel-density-estimation spatial-autocorrelation robust-spatial-regression


Robust Kruskal-Wallis test

Robust Kruskal-Wallis One-Way Analysis of Variance by Ranks

robust K-W test
trimmed Kruskal-Wallis
robust nonparametric one-way test
robust rank-based ANOVA

1952 (base); robust variants 1990s–2000s

Kruskal & Wallis (1952); robust extensions by Wilcox and others

The robust Kruskal-Wallis test is a nonparametric, rank-based method for comparing three or more independent groups when data contain outliers, heavy tails, or heterogeneous spread. It augments the classical Kruskal-Wallis H statistic with robust techniques — such as trimmed means on ranks or permutation-based inference — to maintain valid Type I error rates even when distributional assumptions are violated.

Ordinal or continuous with outliers or heavy-tailed distributions

Mielke, P. W., & Berry, K. J. (2007). Permutation Methods: A Distance Function Approach (2nd ed.). Springer.

kruskal-wallis-test robust-one-way-anova mann-whitney-u-test robust-mann-whitney-u-test friedman-test permutation-kruskal-wallis-test


Robust Latent Class Analysis

trimmed-likelihood latent class analysis

Building on Hennig (2004) and Vermunt & Magidson (2004)

Robust latent class analysis (robust LCA) extends the standard latent class model by incorporating outlier-resistant estimation techniques — such as trimmed likelihood, M-estimation, or downweighting — so that atypical response patterns do not distort the recovered class structure or class membership probabilities.

Categorical (nominal or ordinal) multivariate indicators

Hennig, C. (2004). Breakdown points for maximum likelihood estimators of location-scale mixtures. Annals of Statistics, 32(4), 1313–1340.

Vermunt, J. K., & Magidson, J. (2004). Latent class models. In D. Kaplan (Ed.), The Sage Handbook of Quantitative Methodology for the Social Sciences (pp. 175–198). Sage.

latent-class-analysis mixture-modeling robust-mixture-modeling robust-latent-profile-analysis robust-exploratory-factor-analysis cluster-analysis


Robust Latent Profile Analysis

robust mixture model for continuous indicators

Building on Vermunt & Magidson (2002); robust extensions developed through contaminated normal mixture literature (Punzo & McNicholas, 2010s)

Robust latent profile analysis identifies latent subgroups of individuals based on their continuous multivariate indicators while protecting parameter estimates from distortion by outliers or atypical observations. It extends standard latent profile analysis by replacing the Gaussian component densities with heavier-tailed or contaminated-normal alternatives that down-weight extreme cases during estimation.

Person-centered mixture model with robust estimation

Continuous multivariate indicators, potentially containing outliers

Vermunt, J. K. & Magidson, J. (2002). Latent class cluster analysis. In J. A. Hagenaars & A. L. McCutcheon (Eds.), Applied Latent Class Analysis (pp. 89–106). Cambridge University Press.

Punzo, A. & McNicholas, P. D. (2016). Robust clustering in regression analysis via the contaminated Gaussian cluster-weighted model. Journal of Classification, 33(2), 293–331.

latent-profile-analysis latent-class-analysis mixture-modeling robust-latent-class-analysis robust-mixture-modeling multilevel-latent-profile-analysis


Robust LightGBM

Robust LightGBM (Light Gradient Boosting Machine with Robust Loss Functions)

Robust LGBM
LightGBM with Huber loss
outlier-resistant gradient boosting
robust gradient boosted trees

2017 (LightGBM); robust variants widely adopted 2018–present

Ke, G. et al. (LightGBM); robust objectives adapted from Friedman, J. H.

Robust LightGBM is a gradient boosting framework that pairs Microsoft's highly efficient LightGBM engine with outlier-resistant loss functions — most commonly Huber, quantile, or mean absolute error — so that predictions are not unduly distorted by extreme or erroneous observations. It retains LightGBM's speed and leaf-wise tree growth while providing resistance to heavy-tailed noise in the target variable.

Ensemble (gradient boosted decision trees with robust loss)

Tabular — continuous, categorical, binary targets; datasets with potential outliers

Ke, G., Meng, Q., Finley, T., Wang, T., Chen, W., Ma, W., Ye, Q., & Liu, T.-Y. (2017). LightGBM: A Highly Efficient Gradient Boosting Decision Tree. Advances in Neural Information Processing Systems, 30, 3146–3154.

Friedman, J. H. (2001). Greedy Function Approximation: A Gradient Boosting Machine. The Annals of Statistics, 29(5), 1189–1232.

lightgbm xgboost catboost gradient-boosting random-forest huber-regression


Robust Linear Programming

Robust Linear Programming — Uncertainty-Aware Linear Optimization

RLP
Robust LP
Tractable Robust LP
Uncertainty-Set LP

Ben-Tal, A. and Nemirovski, A.; further developed by Bertsimas, D. and Sim, M.

Robust Linear Programming (RLP) extends classical linear programming to handle uncertainty in problem data — cost coefficients, constraint coefficients, or right-hand sides — by requiring solutions to remain feasible and near-optimal across all realizations of uncertain parameters within a defined uncertainty set. It replaces probabilistic assumptions with worst-case guarantees, making it practical when distributional knowledge is limited.

Numerical coefficients with uncertainty sets (intervals, ellipsoidal, polyhedral)

Bertsimas, D., Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53.

Ben-Tal, A., Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25(1), 1–13.

stochastic-linear-programming robust-mixed-integer-programming robust-multi-objective-optimization deterministic-linear-programming robust-dynamic-programming robust-goal-programming


Robust Linear Regression

Robust Linear Regression (Outlier-Resistant Estimation)

robust regression
M-estimator regression
Huber regression
outlier-resistant regression

Robust linear regression fits a linear model between predictors and a continuous outcome while down-weighting or discarding influential outliers, preventing the few anomalous observations that OLS is famously sensitive to from distorting the entire estimated line. Major variants include Huber regression, iteratively reweighted least squares (IRLS), RANSAC, and Theil-Sen estimation.

Continuous numerical features and a continuous target

Huber, P. J. (1964). Robust Estimation of a Location Parameter. Annals of Mathematical Statistics, 35(1), 73–101.

Rousseeuw, P. J. & Leroy, A. M. (1987). Robust Regression and Outlier Detection. Wiley.

linear-regression-ml regularized-linear-regression support-vector-machine huber-regression quantile-regression lasso-regression


Robust LISA
outlier-resistant LISA
robust local spatial autocorrelation
LISA with robust weights

Anselin (LISA, 1995); robust extensions by Assuncao & Reis and subsequent spatial statisticians

Robust Local Indicators of Spatial Association extend Anselin's LISA framework to handle outliers, extreme values, and spatially heterogeneous populations. By applying outlier-resistant adjustments to the spatial weights or the standardised values, Robust LISA identifies statistically significant local clusters and spatial outliers without the distortions caused by highly influential observations.

Local spatial autocorrelation statistic (robust variant)

Georeferenced areal or point data, potentially with outliers or heterogeneous populations

Anselin, L. (1995). Local indicators of spatial association—LISA. Geographical Analysis, 27(2), 93–115.

Assuncao, R. M., & Reis, E. A. (1999). A new proposal to adjust Moran's I for population density. Statistics in Medicine, 18(16), 2147–2162.

local-indicators-of-spatial-association local-morans-i local-gearys-c local-getis-ord-gi-star robust-spatial-autocorrelation spatial-autocorrelation


Robust Logistic Regression

Robust Logistic Regression (Mallows-Type Weighted Estimation)

robust binary regression
weighted logistic regression
Mallows-type logistic regression
Robust Lojistik Regresyon

Cantoni & Ronchetti (2001); Bondell (2008)

Robust Logistic Regression is a variant of logistic regression that is resistant to outliers and leverage points, fitting a binary or categorical outcome with Mallows-type weighted estimation. The robust framework for generalized linear models was developed by Cantoni and Ronchetti (2001), with a weighting approach later refined by Bondell (2008).

Robust generalized linear model (binary outcome)

Cantoni, E. & Ronchetti, E. (2001). Robust Inference for Generalized Linear Models. Journal of the American Statistical Association, 96(455), 1022-1030.

Bondell, H. D. (2008). Robust Logistic Regression Using a Weighting Approach. Biometrics, 64(2), 421-427.

logistic-regression ols-regression mm-estimator robust-time-series quantile-regression


Robust MA model

robust MA
robust moving average
M-estimation MA
bounded-influence MA

Denby & Martin (1979); Muler, Pena & Yohai (2009)

The Robust MA model applies robust estimation — typically M-estimation or bounded-influence methods — to the Moving Average time series model. By replacing the ordinary least squares loss with a bounded loss function, it produces parameter estimates that are far less sensitive to outliers, additive noise spikes, or heavy-tailed error distributions than the classical Gaussian MA.

Univariate time series with potential outliers or heavy-tailed errors

Denby, L., & Martin, R. D. (1979). Robust estimation of the first-order autoregressive parameter. Journal of the American Statistical Association, 74(365), 140–146.

Muler, N., Pena, D., & Yohai, V. J. (2009). Robust estimation for ARMA models. Annals of Statistics, 37(2), 816–840.

moving-average-model arma-model robust-arma-model robust-arima-model robust-ols arima-model


Robust Mann-Whitney U test

robust nonparametric two-group comparison

Rand Wilcox (robust extensions); original test by Mann & Whitney (1947)

The Robust Mann-Whitney U test is a nonparametric two-group comparison that combines the rank-based logic of the classic Mann-Whitney U test with modern robust techniques — such as outlier screening, trimmed means, or robust variance estimation — to produce reliable inferences when data contain extreme values, heavy-tailed distributions, or other violations that compromise the standard test.

Robust nonparametric two-group comparison

Continuous or ordinal data, possibly with outliers

Wilcox, R. R. (2005). Introduction to Robust Estimation and Hypothesis Testing (2nd ed.). Academic Press.

Wilcox, R. R., & Keselman, H. J. (2003). Modern robust data analysis methods: Measures of central tendency. Psychological Methods, 8(3), 254–274.

mann-whitney-u-test wilcoxon-signed-rank-test robust-independent-samples-t-test kruskal-wallis-test permutation-mann-whitney-u-test bootstrap-mann-whitney-u-test


Robust MANOVA

Robust Multivariate Analysis of Variance

robust multivariate ANOVA
trimmed-mean MANOVA
outlier-resistant MANOVA
robust MANOVA

Rand Wilcox; Lisa Lix and H. J. Keselman

Robust MANOVA is a multivariate analysis of variance procedure designed to remain valid when classical assumptions — multivariate normality and homogeneity of covariance matrices — are violated. It replaces raw means and standard covariance matrices with resistant estimates such as trimmed means and Winsorized covariances, yielding reliable Type I error control and power in the presence of outliers and skewed distributions across multiple dependent variables simultaneously.

Continuous multivariate (2+ dependent variables, 2+ independent groups)

Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

Lix, L. M., & Keselman, H. J. (2004). Multivariate tests of means in independent groups designs: Effects of covariance heterogeneity and nonnormality. Evaluation and the Health Professions, 27(1), 45–69.

manova robust-ancova robust-one-way-anova robust-mancova robust-repeated-measures-anova robust-two-way-anova


Robust Marginal Structural Model

Robust Marginal Structural Model with Stabilized Inverse Probability Weighting

robust MSM
doubly-robust MSM
sandwich-SE MSM
robust IPTW marginal structural model

Robins, Hernán & Brumback; robustness extensions by Scharfstein, Rotnitzky, Lunceford & Davidian

Robust Marginal Structural Models (robust MSMs) extend the standard MSM framework — which uses inverse probability of treatment weighting to handle time-varying confounding — by pairing IPTW estimation with sandwich (robust) standard errors or doubly-robust estimators. This combination yields valid causal estimates and reliable inference even when the outcome regression model is mildly misspecified or weights are moderately variable.

Longitudinal / panel data with time-varying treatment

Robins, J. M., Hernán, M. A., & Brumback, B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology, 11(5), 550-560.

Hernán, M. A., & Robins, J. M. (2020). Causal Inference: What If. Chapman & Hall/CRC.

marginal-structural-model inverse-probability-weighting doubly-robust-estimation propensity-score-weighting difference-in-differences panel-data-marginal-structural-model


Robust Markov chain Monte Carlo

Robust Markov Chain Monte Carlo Sampling

robust MCMC
outlier-robust MCMC
robust posterior sampling
misspecification-robust MCMC

Roberts, Rosenthal and colleagues; extended by Atchade, Barp, Girolami and others

Robust MCMC combines Markov chain Monte Carlo sampling with robustness techniques to produce reliable posterior inference when data contain outliers, when the assumed model is misspecified, or when the target distribution has heavy tails that cause standard samplers to mix poorly or yield distorted estimates.

continuous, heavy-tailed, or contaminated data

Roberts, G. O. & Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probability Surveys, 1, 20–71.

Barp, A., Kennedy, C., Durmus, A. & Girolami, M. (2022). Targeted separation and convergence with kernel discrepancies. arXiv preprint.

mcmc gibbs-sampling metropolis-hastings hamiltonian-monte-carlo robust-bayesian-inference sequential-monte-carlo


Robust Markov Model

Robust Markov Model — Markov chain analysis under transition probability uncertainty

RMM
Robust Markov Chain
Uncertain Markov Model
Interval Markov Model

A Robust Markov Model applies robustness principles to Markov chains by replacing single-point transition probabilities with uncertainty sets, then optimizing against the worst-case realization. Originally developed for robust Markov decision processes in operations research, it is used wherever transition rates are estimated with noise or are subject to adversarial variation, ensuring decisions remain safe across the full uncertainty range.

Transition probabilities, state data, reward/cost matrices

Nilim, A., El Ghaoui, L. (2005). Robust control of Markov decision processes with uncertain transition matrices. Operations Research, 53(5), 780-798.

Iyengar, G. N. (2005). Robust dynamic programming. Mathematics of Operations Research, 30(2), 257-280.

markov-model robust-sensitivity-analysis stochastic-markov-model robust-dynamic-programming scenario-analysis monte-carlo-simulation


Robust Matching Estimator

Bias-Corrected Robust Matching Estimator

bias-corrected matching
Abadie-Imbens matching
AI matching estimator
robust nearest-neighbor matching

The robust matching estimator, developed by Abadie and Imbens (2006, 2011), extends nearest-neighbor matching by adding a regression-based bias correction that removes the finite-sample bias arising when matched units are not perfectly alike. It yields consistent, asymptotically normal estimates of average treatment effects with a heteroskedasticity-robust variance formula that is valid regardless of the number of continuous covariates.

Cross-sectional or panel; continuous or binary outcome; continuous or mixed covariates

Abadie, A., & Imbens, G. W. (2011). Bias-Corrected Matching Estimators for Average Treatment Effects. Journal of Business & Economic Statistics, 29(1), 1-11.

Abadie, A., & Imbens, G. W. (2006). Large Sample Properties of Matching Estimators for Average Treatment Effects. Econometrica, 74(1), 235-267.

propensity-score-matching coarsened-exact-matching matching-estimator doubly-robust-estimation inverse-probability-weighting difference-in-differences


Robust McDonald's Omega

Robust McDonald's Omega Reliability Coefficient

robust omega
omega total (robust)
robust omega-total
robust composite reliability

1999 (omega); robust variant formalized in 2000s–2010s

Roderick P. McDonald (omega); robust extension via robust SEM estimators (MLR, DWLS)

Robust McDonald's omega estimates the internal consistency reliability of a composite scale using factor-analytic loadings obtained through robust estimation methods (such as MLR or DWLS). Unlike standard omega or Cronbach's alpha, it remains accurate when item distributions are non-normal, skewed, or when the sample contains influential outliers — conditions common in applied psychological and educational measurement.

Ordinal or continuous item scores, especially non-normal distributions

McDonald, R. P. (1999). Test theory: A unified treatment. Lawrence Erlbaum Associates.

Dunn, T. J., Baguley, T., & Brunsden, V. (2014). From alpha to omega: A practical solution to the pervasive problem of internal consistency estimation. British Journal of Psychology, 105(3), 399–412.

mcdonalds-omega cronbachs-alpha robust-cronbachs-alpha confirmatory-factor-analysis robust-confirmatory-factor-analysis item-response-theory


Robust Measurement Invariance

robust MI testing
robust measurement equivalence
non-normal measurement invariance
robust multi-group CFA invariance

Robust measurement invariance testing evaluates whether a psychometric instrument measures the same latent construct in the same way across groups when observed data violate multivariate normality. It adapts standard multi-group CFA sequences by replacing ordinary chi-square statistics with robust alternatives such as the Satorra-Bentler scaled statistic, yielding trustworthy conclusions about factor loadings, intercepts, and residual variances even with skewed or ordinal data.

Measurement invariance test with robust corrections

Ordinal or continuous indicators with non-normal distributions

Satorra, A. & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. C. Clogg (Eds.), Latent variables analysis: Applications for developmental research (pp. 399–419). Sage.

Millsap, R. E. (2011). Statistical approaches to measurement invariance. Routledge.

confirmatory-factor-analysis measurement-invariance multigroup-cfa sem partial-measurement-invariance satorra-bentler-chi-square


Robust Mediation Analysis

robust indirect effects
outlier-resistant mediation
robust causal mediation

Yuan & MacKinnon (median-regression formulation, 2014); robust bootstrap variants popularised by Hayes (2013) and Preacher & Hayes (2008)

Robust mediation analysis estimates the indirect effect of an independent variable on an outcome through one or more mediators using estimators that resist the influence of outliers and non-normal error distributions. By combining robust regression (such as median or M-estimation) with percentile or bias-corrected bootstrap confidence intervals, it yields trustworthy conclusions when standard ordinary-least-squares mediation would be distorted by extreme observations.

Continuous or approximately continuous variables (outlier-prone or non-normal distributions)

Yuan, Y., & MacKinnon, D. P. (2014). Robust mediation analysis based on median regression. Psychological Methods, 19(1), 1–20.

Hayes, A. F. (2013). Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach. Guilford Press.

mediation-analysis moderated-mediation robust-structural-equation-modeling robust-path-analysis bootstrap-mediation-analysis structural-equation-modeling


Robust Metric Learning

Robust Metric Learning (Outlier-Resistant Distance Metric Learning)

robust distance metric learning
noise-robust metric learning
outlier-robust similarity learning
robust DML

Various (Weinberger, Saul, Schultz et al.; robust extensions by Shen, Cao and others, 2009–2012)

Robust Metric Learning learns a Mahalanobis distance function from labeled or pairwise-constrained data while actively resisting the distortion caused by noisy labels, corrupted examples, or outliers. By replacing standard hinge or squared losses with robust alternatives and adding regularization, it produces a distance metric that generalises well even when the training set is imperfect — a common situation in real-world scientific and applied tasks.

Supervised/semi-supervised distance metric learning with robustness to noise and outliers

Labeled or pairwise-constrained numerical feature vectors

Shen, C., Kim, J., Wang, L., & van den Hengel, A. (2012). Positive Semidefinite Metric Learning Using Boosting-like Algorithms. Journal of Machine Learning Research, 13, 1007–1036.

Cao, Q., Guo, Z.-C., & Ying, Y. (2012). Generalization Bounds for Metric and Similarity Learning. Machine Learning, 102(1), 115–132.

metric-learning robust-linear-regression robust-support-vector-machine k-nearest-neighbors semi-supervised-metric-learning few-shot-learning


Robust Microsimulation

Robust Microsimulation — Uncertainty-integrated individual-level simulation for policy and health analysis

Robust Micro-Simulation
Uncertainty-Robust Microsimulation
Probabilistic Microsimulation
Sensitivity-Enhanced Microsimulation

Briggs, A. H.; O'Brien, B. J. and others in health technology assessment literature

Robust Microsimulation combines individual-level (micro) simulation with systematic uncertainty analysis — typically probabilistic sensitivity analysis — to generate outputs that are robust to parameter uncertainty, model structure assumptions, and input variability. It is widely used in health technology assessment, public policy, and social science to produce credible, decision-relevant predictions.

Simulation with systematic robustness testing

Individual-level microdata, parameter distributions, probabilistic inputs

O'Brien, B. J., & Briggs, A. H. (2002). Analysis of uncertainty in health care cost-effectiveness studies: an introduction to statistical issues and methods. Statistical Methods in Medical Research, 11(6), 455-468.

Caro, J. J., Briggs, A. H., Siebert, U., & Burgess, K. A. (2012). Modeling good research practices — overview: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-1. Medical Decision Making, 32(5), 667-677.

microsimulation monte-carlo-simulation stochastic-microsimulation deterministic-microsimulation sensitivity-analysis scenario-analysis


Robust Mixed-Integer Programming

Robust Mixed-Integer Programming (RMIP) — Optimization under uncertainty with integer decision variables

RMIP
Robust MIP
Uncertain MIP
Robust MILP/MIQP

Robust Mixed-Integer Programming (RMIP) combines mixed-integer programming with robust optimization to find solutions that remain feasible and near-optimal despite uncertain parameters. Instead of assuming fixed data, it protects decisions against adversarial or worst-case realizations of uncertain inputs, using an explicit uncertainty set to control the degree of conservatism while preserving the combinatorial structure of integer decisions.

Deterministic robust reformulation of MIP under uncertainty

Uncertain parameters with known or estimated uncertainty sets; integer and continuous decision variables

Bertsimas, D., Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53.

Ben-Tal, A., El Ghaoui, L., Nemirovski, A. (2009). Robust Optimization. Princeton University Press, Princeton, NJ.

stochastic-mixed-integer-programming mixed-integer-programming robust-linear-programming robust-multi-objective-optimization stochastic-programming scenario-analysis


Robust Mixed Model

robust mixed-effects model
robust linear mixed model
robust LMM
Robust Karma Etkiler Modeli

Richardson & Welsh (robust REML); Koller (robustlmm implementation)

The robust mixed model is a linear mixed-effects model for panel and repeated-measures data that tolerates outliers and heavy-tailed errors. It replaces the usual likelihood with bounded-influence estimating equations, building on the robust restricted maximum likelihood of Richardson and Welsh (1995) and the robustlmm implementation of Koller (2016).

Robust REML / robust scoring with bounded influence functions

Koller, M. (2016). robustlmm: An R Package for Robust Estimation of Linear Mixed-Effects Models. Journal of Statistical Software, 75(6), 1-24.

Richardson, A. M. & Welsh, A. H. (1995). Robust Restricted Maximum Likelihood in Mixed Linear Models. Biometrics, 51(4), 1429-1439.

linear-mixed-model panel-fixed-effects robust-regression heteroscedasticity-robust-se ols-regression permutation-test


Robust Mixture Modeling

robust mixture model
robust GMM
outlier-robust mixture model
trimmed mixture model

Peel & McLachlan (t-mixture); Garcia-Escudero et al. (trimming framework)

Robust mixture modeling fits finite mixture models — probabilistic clustering methods that assume data arise from a blend of underlying subpopulations — using component distributions or estimation strategies designed to be insensitive to outliers and heavy-tailed noise. The two dominant approaches replace Gaussian components with heavier-tailed distributions such as the multivariate t, or trim a fixed proportion of the most extreme observations before fitting.

Latent-class probabilistic clustering with outlier protection

Continuous multivariate data (possibly with outliers or heavy tails)

Garcia-Escudero, L. A., Gordaliza, A., Matran, C. & Mayo-Iscar, A. (2008). A general trimming approach to robust cluster analysis. Annals of Statistics, 36(3), 1324–1345.

Peel, D. & McLachlan, G. J. (2000). Robust mixture modelling using the t distribution. Statistics and Computing, 10(4), 339–348.

mixture-modeling robust-latent-class-analysis robust-cluster-analysis robust-latent-profile-analysis robust-k-means-clustering robust-principal-component-analysis


Robust Model Testing Research

robust SEM
robust structural model testing
robust fit evaluation
robust model evaluation research

Albert Satorra & Peter M. Bentler; Ke-Hai Yuan

Robust model testing research applies structural or path models to data while explicitly accounting for violations of multivariate normality and other distributional assumptions. Rather than discarding non-normal data or forcing transformations, it uses corrected estimators — most notably the Satorra-Bentler scaled chi-square and Yuan-Bentler robust standard errors — to produce trustworthy fit indices and parameter estimates even when classical maximum likelihood assumptions are breached.

Quantitative model-testing research design with robust estimation

Continuous, ordinal, or non-normal multivariate numeric data

Satorra, A., & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. C. Clogg (Eds.), Latent variables analysis: Applications for developmental research (pp. 399–419). Sage.

Yuan, K.-H., & Bentler, P. M. (1998). Robust mean and covariance structure analysis. British Journal of Mathematical and Statistical Psychology, 51(1), 63–88.

structural-equation-modeling confirmatory-factor-analysis model-testing-research multivariate-model-testing-research bayesian-model-testing-research path-analysis


Robust Moderated Mediation

robust conditional process analysis
robust mediated moderation
robust moderated indirect effects
robust conditional indirect effects

Hayes, A. F.; building on Preacher, Rucker & Hayes (2007) for moderated mediation and robust bootstrap inference

Robust moderated mediation tests whether the indirect effect of X on Y through a mediator M varies as a function of a moderator W, while using robust estimation (percentile or bias-corrected bootstrap, heteroscedasticity-consistent standard errors, or M-estimation) to protect inference against non-normality, outliers, and heteroscedasticity in the data.

Conditional indirect effect model with robust inference

Continuous, ordinal, or mixed; non-normal tolerant

Hayes, A. F. (2022). Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach (3rd ed.). Guilford Press.

Yuan, K.-H., & Bentler, P. M. (2002). On robustness of the normal-theory based asymptotic distributions of three reliability coefficient estimates. Psychometrika, 67(2), 251–259.

moderated-mediation robust-mediation-analysis robust-moderation-analysis structural-equation-modeling bootstrap-mediation-analysis robust-path-analysis


Robust Moderation Analysis

robust interaction analysis
robust moderated regression
HC-corrected moderation
outlier-resistant interaction testing

Robust moderation analysis tests whether the effect of a predictor on an outcome depends on the level of a moderator variable, using estimation methods that remain valid under non-normality, heteroscedasticity, or the presence of influential outliers. It is the preferred approach when standard ordinary least squares assumptions cannot be trusted.

Robust regression-based interaction test

Continuous or ordinal predictors and outcome

Hayes, A. F. & Cai, L. (2007). Using heteroscedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior Research Methods, 39(4), 709–722.

Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

moderation-analysis robust-mediation-analysis robust-path-analysis moderated-mediation robust-structural-equation-modeling robust-multiple-regression


Robust Monte Carlo Simulation

robust MC simulation
Monte Carlo robustness analysis
robust stochastic simulation
uncertainty-robust Monte Carlo

Saltelli, Rubinstein, and the uncertainty-quantification community

Robust Monte Carlo simulation extends standard Monte Carlo by explicitly accounting for uncertainty in input distributions, model structure, or parameter assumptions. Rather than assuming a single fixed probability distribution for each input, the analyst considers a family of plausible distributions and evaluates how sensitive the output is to those choices, yielding conclusions that hold across a range of reasonable assumptions.

Robust simulation / uncertainty quantification

Continuous, distributional, or interval-valued inputs

Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M. & Tarantola, S. (2008). Global Sensitivity Analysis: The Primer. Wiley.

Rubinstein, R. Y. & Kroese, D. P. (2016). Simulation and the Monte Carlo Method (3rd ed.). Wiley.

monte-carlo-simulation robust-bayesian-inference sequential-monte-carlo robust-particle-filter sensitivity-analysis bootstrap-simulation


Robust Moran's I

Robust Moran's I Spatial Autocorrelation Statistic

outlier-resistant Moran's I
robust spatial autocorrelation test
median-based Moran statistic
robust global spatial association

Extension of Moran (1950); robust adaptations developed in spatial statistics literature

Robust Moran's I is an outlier-resistant adaptation of the classic Moran's I spatial autocorrelation statistic. By replacing the standard mean-based standardization with resistant measures of center and spread, it detects genuine geographic clustering without being distorted by a small number of extreme values in the attribute of interest.

Robust spatial autocorrelation statistic

Georeferenced areal or point data with potential outliers

Anselin, L. (1995). Local indicators of spatial association—LISA. Geographical Analysis, 27(2), 93–115.

Lee, J., & Wong, D. W. S. (2001). Statistical Analysis with ArcView GIS. John Wiley & Sons.


Robust Multi-Objective Optimization

Robust Multi-Objective Optimization (RMOO) — optimizing multiple conflicting objectives under uncertainty

Robust Multi-Objective Optimization (RMOO) is a framework for finding solutions that simultaneously optimize multiple conflicting objectives while remaining insensitive to perturbations in decision variables or problem parameters. Unlike classical MOO, RMOO explicitly incorporates uncertainty into the optimization loop, producing a robust Pareto front whose members perform well not only at the nominal design point but also across a neighbourhood of plausible operating conditions.

Continuous or discrete decision variables with uncertain parameters

Deb, K., & Gupta, H. (2006). Introducing robustness in multi-objective optimization. Evolutionary Computation, 14(4), 463–494.

multi-objective-optimization nsga-ii stochastic-multi-objective-optimization robust-optimization pareto-front sensitivity-analysis


Robust Multidimensional Scaling

Robust MDS
outlier-resistant MDS
robust proximity scaling

2002 (robust extension); 1952 (classical MDS)

Hubert, Arabie, and Meulman (robust extensions); classical MDS by Torgerson (1952)

Robust multidimensional scaling recovers a low-dimensional spatial map from a matrix of pairwise dissimilarities while resisting distortion caused by outlying or erroneous proximity values. By replacing squared-error loss with a robust loss function or down-weighting suspect pairs, it produces a configuration that faithfully represents the bulk of the data even when some distances are grossly atypical.

Dimensionality reduction / proximity scaling

Dissimilarity or distance matrices (continuous); tolerates outlying proximities

Hubert, L., Arabie, P. & Meulman, J. (2002). Linear unidimensional scaling in the L2-norm: Basic optimization methods using SMACOF. Journal of Classification, 19(2), 303–327.

Buja, A., Swayne, D. F., Littman, M. L., Dean, N., Hofmann, H. & Chen, L. (2008). Data visualization with multidimensional scaling. Journal of Computational and Graphical Statistics, 17(2), 444–472.

multidimensional-scaling robust-principal-component-analysis robust-cluster-analysis principal-component-analysis robust-exploratory-factor-analysis robust-correspondence-analysis


Robust Multinomial Logistic Regression

robust polychotomous logistic regression

M-estimation multinomial logistic regression

2001 (robust GLM); 1970s–1980s (multinomial logistic regression)

Cantoni & Ronchetti (robust GLM framework); Agresti (multinomial logistic regression)

Robust multinomial logistic regression extends the standard multinomial logit model to handle outliers, influential observations, and mild misspecification of the response distribution. It replaces the conventional maximum likelihood score equations with bounded influence functions (M-estimation) or pairs maximum likelihood with sandwich variance estimators, so that a small fraction of anomalous cases cannot distort the estimated log-odds ratios across outcome categories.

Nominal categorical outcome, continuous or categorical predictors

Cantoni, E., & Ronchetti, E. (2001). Robust inference for generalized linear models. Journal of the American Statistical Association, 96(455), 1022–1030.

Agresti, A. (2002). Categorical Data Analysis (2nd ed.). Wiley-Interscience.

multinomial-logistic-regression robust-logistic-regression ordinal-logistic-regression robust-ordinal-logistic-regression robust-regression generalized-linear-model


Robust Multiple Correspondence Analysis

Robust MCA
Outlier-resistant MCA
Robust HOMALS

Extensions by Hubert, Rousseeuw and collaborators; building on classical MCA by Benzécri (1973) and Greenacre (1984)

Robust Multiple Correspondence Analysis extends classical MCA to datasets containing outlying or atypical rows of categorical data. By downweighting influential observations before the singular value decomposition, it produces a low-dimensional map of category relationships that faithfully represents the bulk of the data rather than being distorted by a handful of anomalous cases.

Multiple categorical (nominal/ordinal) variables

Greenacre, M. J. (2017). Correspondence Analysis in Practice (3rd ed.). Chapman & Hall / CRC Press, Boca Raton.

Hubert, M., Rousseeuw, P. J. & Verboven, S. (2004). A robust PCR method for high-dimensional regressors. Journal of Chemometrics, 17(8–9), 438–452.

multiple-correspondence-analysis correspondence-analysis robust-principal-component-analysis robust-exploratory-factor-analysis principal-component-analysis cluster-analysis


Robust Multiple linear regression

robust MLR
M-estimator regression
resistant multiple regression
robust OLS

Peter J. Huber (M-estimators, 1964); extended by Rousseeuw, Yohai, and Maronna

Robust multiple linear regression estimates the linear relationship between a continuous outcome and several predictors while being resistant to outliers and violations of the normality assumption. Instead of minimising the sum of squared residuals, it uses a bounded loss function — most commonly Huber's or Tukey's bisquare — so that extreme observations receive limited influence on the estimated coefficients.

Continuous outcome, multiple continuous or categorical predictors

Huber, P. J. (1964). Robust estimation of a location parameter. Annals of Mathematical Statistics, 35(1), 73–101.

Maronna, R. A., Martin, R. D., & Yohai, V. J. (2006). Robust Statistics: Theory and Methods. Wiley.

multiple-linear-regression ols-regression quantile-regression robust-regression lasso-regression ridge-regression


Robust Naive Bayes

Naive Credal Classifier
NCC
Robust Bayesian Naive Classifier
Imprecise Naive Bayes

Robust Naive Bayes extends the standard Naive Bayes classifier to handle uncertainty or noise in class-conditional probability estimates by replacing point probability estimates with intervals or sets of distributions. The canonical formulation — the Naive Credal Classifier proposed by Zaffalon (2002) — uses imprecise-probability sets so that predictions are made only when all distributions in the set agree, withholding a label when evidence is insufficient.

Probabilistic generative classifier with imprecise-probability robustness

Categorical, continuous, or mixed tabular features with class labels

Zaffalon, M. (2002). The Naive Credal Classifier. Journal of Statistical Planning and Inference, 105(1), 5–21.

logistic-regression k-nearest-neighbors support-vector-machine gaussian-mixture-model semi-supervised-naive-bayes regularized-naive-bayes


Robust NARDL

Robust Nonlinear Autoregressive Distributed Lag Model

Robust Nonlinear ARDL
Outlier-Robust NARDL
Robust Asymmetric ARDL
R-NARDL

Extension of Shin, Yu & Greenwood-Nimmo (2014) NARDL framework with robust (outlier-resistant) estimation

Robust NARDL marries the asymmetric cointegration framework of Shin, Yu, and Greenwood-Nimmo (2014) with outlier-resistant estimation. It decomposes a regressor into positive and negative partial sums, tests for asymmetric long-run relationships via a bounds test, and replaces the OLS criterion with an M- or MM-estimator to guard against leverage points and additive outliers common in macroeconomic and financial time series.

Nonlinear time-series regression with robust estimation

Time series (levels or first differences, with outliers)

Shin, Y., Yu, B., & Greenwood-Nimmo, M. (2014). Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In W. C. Horrace & R. C. Sickles (Eds.), Festschrift in Honor of Peter Schmidt (pp. 281–314). Springer.

Autoregressive distributed lag. Wikipedia.

nardl ardl-bounds-test quantile-ardl nonlinear-cointegration ols-regression quantile-regression


Robust Negative Binomial Regression

negative binomial regression with robust standard errors

sandwich-corrected negative binomial regression

Robust Negative Binomial Regression models overdispersed count outcomes using the negative binomial distribution while protecting coefficient inference against misspecification of the variance function. It pairs maximum-likelihood estimation of the mean and dispersion parameters with sandwich (Huber-White) standard errors, yielding valid tests even when the assumed variance structure is only approximately correct.

Overdispersed non-negative integer counts

Hilbe, J. M. (2011). Negative Binomial Regression (2nd ed.). Cambridge University Press.

Zeileis, A., Kleiber, C., & Jackman, S. (2008). Regression Models for Count Data in R. Journal of Statistical Software, 27(8), 1–25.

negative-binomial-regression robust-poisson-regression zero-inflated-model poisson-regression robust-regression generalized-linear-model


Robust Nomological Validity

nomological network validity
robust validity testing
nomological validity
RNV

Cronbach & Meehl (seminal framework); later extended by Shadish, Cook, and Campbell

Robust nomological validity evaluates whether a psychological construct relates to theoretically expected variables in the predicted directions, using statistically robust estimation methods that remain trustworthy when distributional assumptions are violated. It tests the construct's place within its nomological network — the web of theoretical relationships that define its meaning.

Validity assessment / construct validation

Latent variable scores, correlations, SEM fit indices

Cronbach, L. J. & Meehl, P. E. (1955). Construct validity in psychological tests. Psychological Bulletin, 52(4), 281–302.

Lawshe, C. H. (1975). A quantitative approach to content validity. Personnel Psychology, 28(4), 563–575.

confirmatory-factor-analysis convergent-validity discriminant-validity structural-equation-modeling construct-validity average-variance-extracted


Robust NSGA-II

Robust Non-dominated Sorting Genetic Algorithm II

Robust NSGA2
NSGA-II under uncertainty
Uncertainty-aware NSGA-II
RNSGA-II

Robust NSGA-II extends the classic NSGA-II evolutionary algorithm to account for parametric uncertainty, finding Pareto-optimal trade-off solutions that remain high-performing even when input parameters deviate from their nominal values. Instead of optimizing objective values at a single point, it evaluates each candidate solution across a range or distribution of uncertainty realizations and selects for robustness alongside Pareto dominance.

Robust evolutionary multi-objective optimization algorithm

Continuous or discrete decision variables; multiple objective functions with uncertain parameters

Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182-197.

Deb, K., & Gupta, H. (2006). Introducing robustness in multi-objective optimization. Evolutionary Computation, 14(4), 463-494.

nsga-ii multi-objective-optimization stochastic-nsga-ii robust-multi-objective-optimization robust-genetic-algorithm multi-objective-genetic-algorithm


Robust OLS

Ordinary Least Squares with Heteroscedasticity-Consistent Standard Errors

HC robust regression
White robust OLS
sandwich estimator OLS
OLS with robust standard errors

Robust OLS applies ordinary least squares to estimate coefficients and then replaces the classical standard errors with heteroscedasticity-consistent (HC) standard errors — commonly called White standard errors. This leaves the point estimates unchanged while yielding valid t-statistics and confidence intervals even when the error variance is not constant across observations.

Cross-sectional or time-series continuous data

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817–838.

Wooldridge, J. M. (2019). Introductory Econometrics: A Modern Approach (7th ed.). Cengage Learning.

ols-regression weighted-least-squares generalized-least-squares quantile-regression robust-gls panel-fixed-effects-model


Robust One-class SVM

Robust OCSVM
Outlier-robust One-Class SVM
Contamination-tolerant OCSVM
Robust novelty detection SVM

Extensions of Scholkopf et al. (1999); robust variants developed in 2000s–2010s

Robust One-Class SVM extends the classic One-Class Support Vector Machine for novelty and anomaly detection by incorporating robustness mechanisms — such as trimmed objectives, robust kernel choices, or contamination-tolerant loss functions — that reduce the influence of heavy-tailed noise or outliers present in the training data, yielding a decision boundary that better represents the true support of the normal class.

Scholkopf, B., Williamson, R., Smola, A., Shawe-Taylor, J., & Platt, J. (1999). Support vector method for novelty detection. Advances in Neural Information Processing Systems (NeurIPS), 12, 582–588.

Liu, Y., Li, Z., & Zhou, C. (2018). Roseq: Robust and efficient one-class SVM for large-scale novelty detection. IEEE Transactions on Neural Networks and Learning Systems, 29(12), 6290–6304.

one-class-svm isolation-forest autoencoder-anomaly-detection robust-support-vector-machine gaussian-mixture-model robust-isolation-forest


Robust one-sample t-test

Robust One-Sample Location Test Using Trimmed Mean

one-sample trimmed mean test
Yuen one-sample test
robust one-sample location test
trimmed mean t-test

Rand R. Wilcox (extending Yuen's trimmed-mean approach)

The robust one-sample t-test replaces the ordinary mean with a trimmed mean and the sample variance with a Winsorized variance to compare a population location against a hypothesized value. It retains the t-test decision framework while sharply reducing sensitivity to outliers and heavy-tailed distributions, making it reliable in real-world continuous data that deviate from normality.

Continuous (possibly skewed or heavy-tailed)

Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61(1), 165–170.

one-sample-t-test robust-independent-samples-t-test robust-paired-samples-t-test wilcoxon-signed-rank-test sign-test bootstrap-one-sample-t-test


Robust one-way ANOVA

trimmed-mean ANOVA
Welch one-way ANOVA
heteroscedastic one-way ANOVA
robust ANOVA

1951 (Welch); 1990s–2000s (trimmed-mean variants)

B. L. Welch; R. R. Wilcox (trimmed-mean extension)

Robust one-way ANOVA compares the central tendency of three or more independent groups while resisting the distorting effects of outliers and heterogeneous variances. By replacing ordinary means with trimmed means and ordinary variances with Winsorized variances, it maintains accurate Type I error control and strong power when classical ANOVA assumptions are violated.

Continuous outcome, categorical grouping (3+ groups)

Welch, B. L. (1951). On the comparison of several mean values: an alternative approach. Biometrika, 38(3/4), 330–336.

one-way-anova welch-corrected-one-way-anova kruskal-wallis-test robust-independent-samples-t-test trimmed-mean-one-way-anova bootstrap-one-way-anova


Robust Online Learning

Robust Online Learning (Adversarially and Noise-Resilient Sequential Learning)

ROL
robust incremental learning
adversarially robust online learning
robust sequential learning

Hazan, E.; Shalev-Shwartz, S.; and others

Robust Online Learning extends the online learning framework — where a model updates sequentially after each observation — by incorporating robustness mechanisms that guard against corrupted labels, adversarial examples, heavy-tailed noise, and concept drift. The result is a sequential learner that maintains bounded regret even when the data stream contains outliers or deliberate perturbations.

Sequential / streaming tabular data, possibly corrupted or adversarial

Hazan, E. (2016). Introduction to Online Convex Optimization. Foundations and Trends in Optimization, 2(3–4), 157–325.

Shalev-Shwartz, S. (2012). Online Learning and Online Convex Optimization. Foundations and Trends in Machine Learning, 4(2), 107–194.

online-learning robust-gradient-boosting semi-supervised-online-learning online-gradient-descent robust-support-vector-machine active-learning


Robust Optimization

Robust Optimization (Minimax Programming)

Gürbüz Optimizasyon (Robust Optimization)

1970s theoretical roots; modern tractable form from late 1990s–2004

Ben-Tal, El Ghaoui & Nemirovski (seminal book, 2009); Bertsimas & Sim (tractable polyhedral formulation, 2004)

Robust optimization is a mathematical programming framework, formalised by Ben-Tal and Nemirovski in the late 1990s and made broadly tractable by Bertsimas and Sim (2004), that finds decisions guaranteed to perform acceptably under every scenario within a predefined uncertainty set — rather than assuming parameter values are known exactly. Instead of optimising for a single expected outcome, it minimises the worst-case objective across all plausible realisations of uncertain data.

Minimax (worst-case) over an uncertainty set

Ellipsoidal, box, polyhedral, or budget-constrained

Convex reformulation or Benders decomposition

Output: A single robust (worst-case-optimal) decision vector

Ben-Tal, A., El Ghaoui, L. & Nemirovski, A. (2009). Robust Optimization. Princeton University Press.

Bertsimas, D. & Sim, M. (2004). The Price of Robustness. Operations Research, 52(1), 35-53.

stochastic-optimization surrogate-optimization evolutionary-strategy linear-programming convex-optimization


Robust paired samples t-test

trimmed-mean paired t-test
Yuen paired t-test
robust dependent-samples t-test
trimmed paired comparison

The robust paired samples t-test replaces arithmetic means with trimmed means and Winsorized variance to compare two related measurements while resisting the distorting influence of outliers and non-normal distributions, producing reliable inference where the classic paired t-test breaks down.

paired-samples-t-test robust-independent-samples-t-test wilcoxon-signed-rank-test trimmed-mean-paired-samples-t-test bootstrap-paired-samples-t-test robust-one-sample-t-test


Robust Panel Data Analysis

Robust Panel Data Analysis with Cluster-Robust and Heteroscedasticity-Consistent Inference

panel regression with robust standard errors

Arellano (1987); White (1980) heteroscedasticity-consistent framework

Robust panel data analysis applies standard panel estimators — fixed effects, random effects, or pooled OLS — while replacing conventional standard errors with cluster-robust or heteroscedasticity-consistent (HC) variants. The point estimates remain unchanged; what changes is the variance-covariance matrix used for inference, making t-tests and F-tests valid even when errors are heteroscedastic or correlated within cross-sectional units over time.

Robust estimation / inference correction

Balanced or unbalanced panel (cross-sectional time series)

Arellano, M. (1987). Computing robust standard errors for within-groups estimators. Oxford Bulletin of Economics and Statistics, 49(4), 431–434.

Cameron, A. C., & Trivedi, P. K. (2015). Microeconometrics: Methods and Applications. Cambridge University Press.

panel-fixed-effects-model panel-random-effects-model panel-data-analysis fixed-effects-model panel-hausman-test robust-ols


Robust Panel Event Study

heteroskedasticity-robust panel event study

Sun & Abraham (2021); Freyaldenhoven, Hansen, Shapiro & Weidner (2021)

A robust panel event study extends the standard panel event study design by applying heteroskedasticity- and autocorrelation-robust (HAC) standard errors and, where staggered treatment adoption exists, interaction-weighted estimators that remain valid even when treatment effects are heterogeneous across cohorts and time periods. It is widely used in economics, finance, and policy research to trace the dynamic causal path of an intervention.

Longitudinal / panel data with a discrete treatment event

Sun, L., & Abraham, S. (2021). Estimating dynamic treatment effects in event studies with heterogeneous treatment effects. Journal of Econometrics, 225(2), 175-199.

Freyaldenhoven, S., Hansen, C., Shapiro, J. M., & Weidner, M. (2021). Visualization, Identification, and Estimation in the Linear Panel Event-Study Design. NBER Working Paper No. 29170.

panel-event-study dynamic-difference-in-differences difference-in-differences staggered-difference-in-differences event-study-design panel-fixed-effects


Robust Particle Filter

RPF
robust sequential Monte Carlo
outlier-robust particle filter
heavy-tailed particle filter

Hurzeler & Kunsch; Ristic, Arulampalam & Gordon

The Robust Particle Filter is a sequential Monte Carlo method that tracks hidden states in nonlinear, non-Gaussian systems while remaining resistant to outliers and model misspecification. It replaces the standard Gaussian likelihood with a heavy-tailed or bounded-influence density, so that anomalous observations receive downweighted importance and cannot derail the state estimate.

Ristic, B., Arulampalam, S. & Gordon, N. (2004). Beyond the Kalman Filter: Particle Filters for Tracking Applications. Artech House.

Hurzeler, M. & Kunsch, H. R. (1998). Monte Carlo approximations for general state-space models. Journal of Computational and Graphical Statistics, 7(2), 175-193.

particle-filter kalman-filter robust-kalman-filter sequential-monte-carlo robust-sequential-monte-carlo hamiltonian-monte-carlo


Robust Particle Swarm Optimization

Robust Particle Swarm Optimization — Uncertainty-aware swarm-based metaheuristic

Robust PSO
RPSO
Uncertainty-robust PSO
PSO with robustness

Kennedy, J. & Eberhart, R. C. (PSO); robustness extensions by multiple authors, 2000s

Robust Particle Swarm Optimization (Robust PSO) extends the classical PSO metaheuristic to explicitly account for uncertainty in the objective function, constraints, or decision variables. Rather than optimizing a single nominal objective, each candidate solution is evaluated over a set of uncertainty scenarios, and fitness is judged by a robustness criterion such as worst-case performance or expected value, yielding solutions that remain near-optimal even when conditions deviate from nominal assumptions.

Metaheuristic — robust swarm-based optimizer

Continuous or mixed decision variables with uncertain parameters or noisy objective evaluations

Kennedy, J., Eberhart, R. C., & Shi, Y. (2001). Swarm Intelligence. Morgan Kaufmann Publishers.

Dellino, G., Kleijnen, J. P. C., & Meloni, C. (2010). Robust optimization in simulation: Taguchi and Response Surface Methodology. International Journal of Production Economics, 125(1), 52–59.


Robust Path Analysis

path analysis with robust standard errors

Yuan & Bentler (robust SEM/path framework); Huber (M-estimation foundation)

Robust path analysis applies robust estimation — such as sandwich standard errors or M-estimation — to path models that specify directed causal relationships among observed variables. It preserves valid inference about path coefficients and indirect effects when data violate normality, contain outliers, or exhibit heteroscedasticity that would distort conventional standard errors.

Causal path modeling with robust estimation

Continuous or ordinal observed variables; non-normal or outlier-prone data

Yuan, K.-H. & Bentler, P. M. (1998). Robust mean and covariance structure analysis. British Journal of Mathematical and Statistical Psychology, 51(1), 63–88.

Hair, J. F., Black, W. C., Babin, B. J. & Anderson, R. E. (2019). Multivariate Data Analysis (8th ed.). Cengage Learning.

path-analysis structural-equation-modeling robust-structural-equation-modeling robust-mediation-analysis mediation-analysis robust-confirmatory-factor-analysis


Robust PCA

RPCA
robust principal component analysis
low-rank plus sparse decomposition
Robust Temel Bileşen Analizi (RPCA)

Candès, Li, Ma & Wright (2011); Hubert, Rousseeuw & Vanden Branden (2005)

Robust Principal Component Analysis is a dimensionality-reduction method that extracts reliable components when the data are contaminated by outliers and noise. Introduced by Candès, Li, Ma and Wright (2011), and developed in the ROBPCA approach of Hubert, Rousseeuw and Vanden Branden (2005), it separates a data matrix into a clean low-rank part and a sparse outlier part.

Robust dimensionality reduction / matrix decomposition

Low-rank plus sparse matrix decomposition

Candès, E. J., Li, X., Ma, Y., & Wright, J. (2011). Robust Principal Component Analysis? Journal of the ACM, 58(3), 1-37.

Hubert, M., Rousseeuw, P. J., & Vanden Branden, K. (2005). ROBPCA: A New Approach to Robust Principal Component Analysis. Technometrics, 47(1), 64-79.

pca robust-covariance-mcd robust-mahalanobis-distance factor-analysis robust-regression