A Comparison of the Most Stringent and the Most Stringent Somewhere Most Powerful Test for Certain Problems with Restricted Alternative
0. Summary. The paper studies hypothesis testing problems (H, K1) for the mean of a vector variate having a multivariate normal distribution with known covariance matrix, in cases where the alternative K1 is restricted by a number of linear inequalities (Section 2). We describe a method for obtaining the most stringent size-a test p* for Problem (H, K1) (Section 3). This method is used for constructing so* for a number of special problems (H, K1) (Sections 5, 6 and 7) where (i) K1' is symmetrical and (ii) 0o is sufficiently small. Thus for these special problems, we can compare sp* with the most stringent SMP size-a test po that can be obtained by applying the general methods of [91 and [10] (described in Section 2). It turns out (for the special problems with a = .05 or .01) that so* does not provide a worth-while improvement upon (Po: so* has a smaller maximum shortcoming but poO is better than s* 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-a test for a very general problem (H, K(m)) where the hypothesis H may be composite while K( (m) consists of afinite number (m) of parameters Oi(i = 1, , m) (Section 4)