Hypotheses Testing for the Common Mean and For Balanced Incomplete Blocks Designs

Abstract

Let $X_1, X_2,\cdots, X_m$ be a random sample of size $m$ from a normal population with mean $\theta$ and variance $\sigma_x^2$. Let $Y_1, Y_2,\cdots, Y_n$ be a random sample of size $n$ from a normal population with mean $\theta$ and variance $\sigma_y^2$. The $X$-sample and $Y$-sample are independent. Note that this model is appropriate in balanced incomplete blocks designs. We consider various hypotheses testing problems concerned with $\theta$ and obtain the following results: (1) For testing $H_0: \theta = 0 \mathrm{vs} H_1: \theta \neq 0$ (and the usual variants of these hypotheses), the usual $t$-test based on only one sample is proven to be admissible. This is somewhat surprising in light of results obtained in point and confidence estimation. (2) For testing $H_0: \theta = 0 \mathrm{vs} H_1 \neq 0$, suppose it is assumed that $\sigma_x \geqq \mathbf{B}$, where $\mathbf{B}$ is any positive constant. Then a similar test is found, which is better than the $t$-test based only on the $X$-sample, if $m \geqq 3$ and $n \geqq 4$. (3) Let $\delta = \theta/\sigma_x$ and test $H_0: \delta = 0 \mathrm{vs} H_1: \delta_0 \leqq \delta \leqq \delta_1$. Here $\delta_0 > 0$ can be determined by the sample sizes and size of the $t$-test and $\delta_1$ is arbitrarily large. For this separated hypothesis a test, based on improved estimators of $\theta$, is found which is better than the usual $t$-test for $m \geqq 2, n \geqq 6$. (4) Let $\sigma_x^2$ be known and test $H_0: \theta = 0 \mathrm{vs} H_1 \theta \neq 0$. It is shown in this case that the test which rejects if $|\bar{X}| > C$, is admissible.