### Hsu's Work on Inference

E. L. Lehmann
1979 Annals of Statistics
The first of these [2] is concerned with what today is called the Behrens-Fisher problem. If X; and lj(i = I, · · · , m;j = 1, ... , n) denote samples from normal distributions N(~, o 2 ) and N(TJ, T 2 ), Hsu considers the class of statistics u = (Y-X) 2 1(A 1 S1 + A 2 S~) where s; =~(X; Xf and Si = ~( lJ -:Y)2. This reduces to u 1 , Student's t, for A 1 = A 2 = N I mn(N -2) where N = m + n and to the Behrens-Fisher statistic u 2 for A 1 = 1 I m(m -I), A 2 = Iln(n-1). Hsu finds a series
more » ... ds a series expansion for the density of u, and utilizes this to study the power function of the rejection regions u > C in terms of the parameters (} = T 2 I o 2 2 2 and A. = ( 11 -~2 I ( !'__ + .!__ ). It is an exact (not asymptotic) analysis, described by m n Scheffe (1970) as "a model of mathematical rigor". In the process, he obtains stochastic bounds for u 2 which were later taken up independently and generalized by Hajek, Lawton and others (cf. Eaton and Olshen (1972) ). Hsu's main conclusion, obtained by a combination of his analytical study with some numerical work, is that for A. = 0 and varying(} neither u 1 nor u 2 control the rejection probability at all well (except when m = n) although of the two, u 2 is less sensitive to variation of 9. In the second paper [3], Hsu treats the question of optimal estimators of the variance o 2 in the Gauss-Markov model. In the spirit of the Gauss-Markov theorem, he considers estimators Q which are (a) quadratic and (b) unbiased. In addition he imposes the restriction (c) that the variance of Q be independent of the unknown means. (This is a forerunner of the condition he imposed in [12] for the power function of analysis of variance tests). Hsu then obtains a necessary and sufficient condition for the usual unbiased estimate S 2 of o 2 to have uniformly minimum variance within this class of estimators. He illustrates the condition on a number of examples and, in particular, shows that S 2 has the desired property in the one-sample case. The problem was