Some two Sample Tests

Abstract

Stein [4] has exhibited a double sampling procedure to test hypotheses concerning the mean of normal variables with power independent of the unknown variances. This procedure is here adapted to test hypotheses concerning the ratio of means of two normal populations, also with power independent of the unknown variances. The use of a two sample procedure in a regression problem is also considered. Let $\{X_{ij}\} (i = 1, 2) (j = 1, 2, 3, \cdots)$ be independent random variables distributed according to $N(m_i, \sigma_i):$ all parameters are assumed to be unknown. Defining $k$ by the equation \begin{equation*}\tag{(1)} m_1 = km_2\end{equation*} we wish to test the hypothesis $H$ that $k$ has a specified value $k_0$. If $k_0 = 1$ the hypothesis $H$ reduces to a classical problem, often referred to in the literature as the Behrens-Fisher-problem (cf. Scheffe [3] for a bibliography). At the present time it is still an open question whether it is possible (or desirable) to find a non-trivial single sample test for $H$ with the size of the critical region independent of $\sigma_1$ and $\sigma_2$. In any case it is a simple extension of the result of Dantzig [1] (cf. also Stein [4]) to show that no non-trivial single sample test exists whose power is independent of $\sigma_1$ and $\sigma_2$. On the other hand the case $k_0 \neq 1$ may be expected to occur frequently in fields of application where a choice must be made between different products, methods of experimentation etc. which involve different costs. The statistician must make a choice on the basis of results relative to the ratio of costs involved. Nevertheless this problem appears to have received little attention in the literature. In general tests based on a two-sample procedure may not be as "efficient" in the sense of Wald [5] as a strict sequential procedure. On the other hand the two sample procedure reduces the number of decisions to be made by the experimenter and it will, in certain fields, simplify the experimental procedure.