On the Power of Optimum Tolerance Regions when Sampling from Normal Distributions

Abstract

In [1], optimum $\beta$-expectation tolerance regions were found by reducing the problem to that of solving an equivalent hypothesis testing problem. The regions produced when sampling from a $k$-variate normal distribution were found to be of similar $\beta$-expectation and optimum in the sense of minimax and most stringency. It is the purpose of this paper to discuss the "Power" or "Merit" of such regions, when sampling from the $k$-variate normal distribution. Let $X = (X_1, \cdots, X_n)$ be a random sample point in $n$ dimensions, where each $X_i$, is an independent observation, distributed by $N(\mu, \sigma^2)$. It is often desirable to estimate on the basis of such a sample point a region which contains a given fraction $\beta$ of the parent distribution. We usually seek to estimate the center 100$\beta$% of the parent distribution and/or the 100$\beta$% left-hand tail of the parent distribution.