INADMISSIBILITY OF STUDENTIZED TESTS FOR NORMAL ORDER RESTRICTED MODELS

Consider the model where X(ij), i = 1,...,k; j = 1,2,...,n(i); n(i) greater-than-or-equal-to 2, are observed. Here X(ij) are independent N(theta(i), sigma2), theta(i) sigma2 unknown. Let X(i) = SIGMA(j=1)n X(ij/n(i), X' = (X1,...,X(k)), theta' = (theta1,...,theta(k)), V = SIGMA(i=1)k SIGMA(j=1)i X(ij)2 - nSIGMA(i=1)k X(i)2. Let A1 be a (k-m) x k matrix of rank (k-m) greater-than-or-equal-to 2 and test H:A1theta=0 versus K-H where K:A1theta greater-than-or-equal-to 0. Suppose we assume sigma2 known and consider a constant size alpha test (alpha < 1/2) which is admissible for H versus K-H based on X. Next assume sigma2 is unknown. Consider the same test but now as a function of X/V1/2 (i.e., Studentize the test). The resulting test is inadmissible. Examples are noted.