Confidence Intervals for the Normal Mean with Known Coefficient of Variation

In this paper we proposed two new confidence intervals for the normal population mean with known coefficient of variation. This situation occurs normally in environment and agriculture experiments where the scientist knows the coefficient of variation of their experiments. We propose two new confidence intervals for this problem based on the recent work of Searls [5] and the new method proposed in this paper for the first time. We derive analytic expressions for the coverage probability and the expected length of each confidence interval. Monte Carlo simulation will be used to assess the performance of these intervals based on their expected lengths. Keywords—confidence interval, coverage probability, expected length, known coefficient of variation.


I. INTRODUCTION
T He problem of inference concerning the confidence interval for the normal population mean has been studied recently. In the routine text such as Walpole et al. [7] shows that the confidence intervals for the normal means can be constructed in two cases; a) variance is known b) variance is unknown. However, in practice, there are situations in area of agricultural, biological, environmental and physical sciences that a coefficients of variation are known. For example, in environmental studies, Bhat and Rao [1] argued that there are some situations that show the standard deviation of a pollutant is directly related to the mean, that means the τ is known. Furthermore in clinical chemistry, " when the batches of some substance (chemicals) are to be analyzed, if sufficient batches of the substances are analyzed, their coefficients of variation will be known". Brazauskas and Ghorai [2] also gave some examples in medical, biological and chemical experiments shown that in practice there are problems concerning that coefficients of variation are known. Most of this statistical problem is due to the estimation of the mean of normal distribution with known coefficient of variation see e.g. Khan [3] , Searls ([5], [6]) and the references cited in the mentioned papers. Recent paper of Bhat and Rao [1] extended the mentioned papers to the test for the normal mean, when its coefficient of variation is known. Their simulation results shown that, for the two-sided alternatives the likelihood ratio test or the Wald test is the best test. This paper extends the recent work of Bhat and Rao [1] to the confidence interval for the normal population mean with known coefficient of variation. We propose two new confidence intervals based on the estimator of the mean with known coefficient of variation of Searls [5] where he showed that the mean squares error of his proposed Suparat Niwitpong is with the Department of Applied Statistics, King Mongkut's University of Technology North Bangkok, Bangkok, THAILAND, 10800, e-mail: suparatn@kmutnb.ac.th estimator is lower than the unbiased estimator (the sample mean). We construct this confidence interval using the central limit theorem. The other confidence interval is constructed base on the relation between the coefficient of variation and the population mean and it standard deviation that will show in the next section. We assess these two new confidence intervals using the coverage probability and the ratio of expected lengths between confidence intervals. Typically, we prefer confidence interval with coverage probability at least the nominal value (1 − α) and its expected length is short.

II. CONFIDENCE INTERVALS FOR NORMAL POPULATION
MEAN Let X = [X 1 , . . . , X n ] be a random sample from the normal distribution with mean μ and standard deviation σ. The sample mean and variance for X are, respectively, denoted asX and We are interested in 100(1-α)% confidence interval for μ. 1) Standard confidence interval for μ: In practice, σ 2 is an unknown parameter and S 2 is an unbiased estimator for σ 2 . In this case, a well-known 100(1 − α)% confidence for μ is where c is t 1−α/2 , an upper 1 − α/2 percentiles of the tdistribution with n-1 degrees of freedom. In the next section we review a method to construct a confidence interval for μ with known coefficient of variation.

MEAN WITH KNOWN COEFFICIENT OF VARIATION
It is known that the unbiased estimator for μ is the mean X, however, when we known a prior information, for example the coefficient of variation (τ = σ/μ), the estimatorX is not appropriated. Searls [5] proposed the estimatorX * = (n + τ 2 ) −1 n i=1 X i where he showed that this estimator has lower mean squares error than that of the unbiased estimator. Further, it is easy to show that the variance of the estimatorX * is where d is z 1−α/2 , an upper 1−α/2 percentiles of the standard normal-distribution. We now propose the new confidence interval for μ based on a prior information τ = σ/μ. It is easy to see that μ = σ/τ , hence our proposed confidence interval for μ is equal to the confidence interval of σ/τ which is where u ∼ χ 2 (n−1),(1−α/2) and v ∼ χ 2 (n−1),α/2 and u and v are an upper 1 − α/2 percentiles and lower α/2 percentiles of the Chi-squared distribution.
Note that confidence intervals (2) and (3) have a priori information τ in their confidence intervals whereas confidence interval (1) has no a priori information in its confidence interval.

IV. COVERAGE PROBABILITIES AND EXPECTED LENGTH OF EACH CONFIDENCE INTERVAL FOR MEAN
2) Coverage probability and Expected length of CI μ : Following Niwitpong and Niwitpong [4], we now derive an analytic expression for the coverage probability for CI μ . Let P (μ ∈ CI μ ) be the coverage probability of confidence interval CI μ then an analytic expression for this confidence coverage is given by Z is the standard normal distribution and Φ(·) is a standard normal function. The expected length of CI μ is therefore 3) Coverage probability and Expected length of CI s : Similarly to CI μ , we now derive an analytic expression for the coverage probability for CI s . Let P (μ ∈ CI s ) be the coverage probability of confidence interval CI s then an analytic expression for this confidence coverage is given by

4) Coverage probability and Expected length of CI p :
Similarly to CI μ , we now derive an analytic expression for the coverage probability for CI p . Let P (μ ∈ CI p ) be the coverage probability of confidence interval CI p then an analytic expression for this confidence coverage is given by The expected length of CI p is therefore

V. SIMULATION FRAMEWORK
In this section, since we have proved that the coverage probabilities of all confidences intervals are equal to 1 − α, we therefore only use Monte Carlo simulation to assess three confidence intervals notified in the previous section: CI μ , CI s and CI p based on their average length widths. We design a simulation, without losing generality, by setting σ = 1, τ = 0, 0.01, 0.05, 0.5, 0.1, 0.3, 0.5, 0.8, 0.9, 1, 1.1, 1.3, 1.5, 1.9, 2.5 and the samples sizes n = 10, 30, 50, 100. We wrote function in R program to generate the data which is normally distributed with means and variances which are mentioned previously to construct three confidence intervals, i.e. CI μ , CI s and CI p and then compute average length width of each confidence interval. All results are illustrated in Table I.

VI. SIMULATION RESULTS
From Table I, we found that the expected length the CI μ when known coefficient of variation is wider than those two new confidence intervals for every sample sizes. For known coefficient of variation (τ ) and small sample sizes i.e. n = 10, 30, the confidence interval CI s is shorter than the confidence interval CI p when τ ≤ 0.5, otherwise the confidence interval CI p is preferable. For n = 50, 100, the confidence interval CI p is shorter than that confidence interval CI s when τ ≥ 0.8.

VII. CONCLUSION
In this paper we proposed new confidence intervals for the normal population mean with known coefficient of variation. We derived, mathematically, coverage probabilities and expected lengths of these intervals. It is shown in sections IV that the coverage probabilities of CI μ and CI s and CI p are equal to the nominal value 1 − α. To compare confidence intervals between CI μ and CI s and CI p , one can compare the expected lengths of each interval derived in sections IV. It is recommended that for known small coefficient of variation i.e. τ ≤ 0.5, the confidence interval CI s is preferable otherwise we choose the confidence interval CI p . Further research may consider statistical estimation for the difference between normal means and normal variances with known coefficients of variation. Also one might consider to derive their coverage probabilities and expected lengths as shown in this paper.

ACKNOWLEDGMENT
The author is grateful to King Mongkut's University of Technology North Bangkok for their partial finance support.