S Two-sample problem in the Gaussian product model. Let X1; : : : ; Xk, X0 1; : : : ; X0 l be independent random variables with distribution Nm;v and Nm0;v , respectively; the parameters m;m0 and v are assumed to be unknown. Show that each likelihood ratio test for the test problem H0 W m m0 against H1 W m > m0 has a rejection region of the form

{M-M'>c(+)V*} with MX, M' = += X;, and = V* = k +1-2 ((X-M) + (x; -M')). i=1