On Hodges-Lehmann optimality of LR tests

It is shown that the likelihood ratio test statistics are Hodges-Lehmann optimal for testing the null hypothesis against the whole parameter space, provided that certain regularity conditions are fulfilled. These conditions are verified for the non-singular normal, multinomial and Poisson distribution. Let {F 7 ; 7 € S } be a family of probability measures, defined on (X, T) by means of the densities {f(x, 7); 7 G E } with respect to a measure v. If we denote the g-fold products defined on the cr-algebra S, describes independent sampling from the q populations Throughout the paper we shall assume that Mfloce. In describing asymptotic properties of tests of the null hypothesis we shall use the notation V = < p€R q ; ^2PJ = 1 and Pj > 0 for all j \ (4)