Possibilistic Hypothesis Testing: A Specificity–Entailment Framework for Confidence Bounds on Natural Possibility Distributions

Abstract: Specificity and entailment are the two governing principles of possibility theory, yet no existing formulation – including our own frequency-based specificity measure. 𝑺𝒑(𝑿) – translates them into a procedure for testing a candidate value against observed data with a controlled error rate, or into a confidence interval in the statistical sense. This gap motivates the present study: a decision-maker using a possibility distribution has no principled way to state how confident that distribution’s dominant category actually is. We close this gap by formulating a possibilistic analogue of classical null-hypothesis testing, built entirely from the natural possibility and natural probability measures of an explanatory database. We define Type I and Type II error functions 𝜶(𝝉) and 𝜷(𝝉) as, respectively, the entailment risk and the specificity risk of an 𝜶-cut acceptance region, and prove that inverting the resulting test yields a confidence interval 𝑪𝑰𝟏−𝜶 that is exactly an 𝜶-cut of the possibility distribution – a result that gives Yager’s classical specificity integral a genuine statistical interpretation it previously lacked. We further introduce the Specificity–Entailment Operating Characteristic (SEOC), a possibilistic analogue of the Receiver Operating Characteristic (ROC) curve, with proofs of its monotonicity and its relationship to specificity, and a two-sample permutation test that replaces a previously heuristic criterion for merging database-held observations with a thinly sampled observation set. The framework is validated on two independently sourced real datasets – India Meteorological Department district rainfall-category counts for Chhattisgarh, and an institutional grade distribution for a university literature course – through a 𝟐𝟎𝟎𝟎-resample coverage simulation, comparison against classical Wilson-score intervals, and an augmentation-test experiment. The coverage simulation shows that 𝑪𝑰𝟏−𝜶 attains conservative, over-nominal coverage under discrete possibility distributions, and reveals that bootstrap and delta-method critical values diverge once a single category’s share exceeds roughly one half – a finding that revises our own initial expectation and indicates the bootstrap procedure should be preferred whenever modal dominance is strong. Together, these results give researchers and practitioners working with possibility distributions a tested, statistically grounded procedure for deciding how much a distribution’s dominant category can be trusted, and how much data is required before that decision is reliable.