A Comparison of the Most Stringent and the Most Stringent Somewhere Most Powerful Test for Certain Problems with Restricted Alternative

Abstract

The paper studies hypothesis testing problems $(H, K_1)$ for the mean of a vector variate having a multivariate normal distribution with known covariance matrix, in cases where the alternative $K_1$ is restricted by a number of linear inequalities (Section 2). We describe a method for obtaining the most stringent size-$\alpha$ test $\varphi^\ast$ for Problem $(H, K_1)$ (Section 3). This method is used for constructing $\varphi^\ast$ for a number of special problems $(H, K_1)$ (Sections 5, 6 and 7) where (i) $K'_1$ is symmetrical and (ii) $\psi_0$ is sufficiently small. Thus for these special problems, we can compare $\varphi^\ast$ with the most stringent SMP size-$\alpha$ test $\varphi_0$ that can be obtained by applying the general methods of [9] and [10] (described in Section 2). It turns out (for the special problems with $\alpha = .05$ or .01) that $\varphi^\ast$ does not provide a worth-while improvement upon $\varphi_0 : \varphi^\ast$ has a smaller maximum shortcoming but $\varphi_0$ is better than $\varphi^\ast$ from other over-all points of view. This supports the opinion that generally no serious objections can be made to the use of the MSSMP tests constructed in [10] Part 2 for problems from actual practice. This is a fortunate circumstance, for these MSSMP tests require only simple calculations. By the way we prove a theorem characterizing the MS size-$\alpha$ test for a very general problem $(H, K_{(m)})$ where the hypothesis $H$ may be composite while $K_{(m)})$ consists of a finite number $(m)$ of parameters $\theta_i(i = 1,\cdots, m)$ (Section 4).