Relationship Between the Fisher Index of Discrimination and the Minimum Test Sample Size

Abstract

In this paper, we study the relationship between the Fisher index of discrimination of a univariate test statistic and the minimum sample size corresponding to the values of the parameters under testing, which is required to achieve predetermined probabilities of the Type I and Type II errors. We present a numerical study of the Fisher indices of discrimination of a gamma statistic and a Poisson statistic used to discriminate between the variances of a normal distribution. For fixed probabilities of the Type I and Type II errors, we show that the Fisher indices of these two statistics converge to some constant value associated with the Fisher index of a certain normal statistic, as the minimum sample size required to separate the two hypotheses goes to infinity, that is, when the two variances under testing become identical. To discriminate between two given variances of a normal distribution, approximate formulae for determining the minimum sample size required to achieve predetermined probabilities of the Type I and Type II errors are derived.