Bahadur Efficiencies of the Student's $t$-Tests

Abstract

An extremal problem in large deviation is solved for the one and two sample $t$-statistic. Let $T_n = \bar{X}/s_X$ be the $t$-statistic, except for normalizing constants, based on $n$ independent observations fron $F$ where $\bar{X}$ and $s_X$ are the sample mean and standard deviation. The statistic for the two sample case is $T_n = (\bar{X} - \bar{Y})/\lbrack(s^2_X + s^2_Y)/2\rbrack^\frac{1}{2}$ based on two independent samples of size $n$. Let $a$ be positive. We find the rate of convergence to 0 of $\sup_F P_F(T_n \geqq a)$ where the $\sup$ is taken over $F$'s symmetric at 0 in the one sample case and all $F$'s in the two sample case. The results are applied to obtain the Bahadur exact slopes of the $t$-statistic for the full nonparametric hypothesis testing problems. We include a table giving the slopes of the $t$-statistics, the slopes of standard rank statistics and the maximum possible slopes. This table shows that, in both the one and two sample situations, the $t$-statistic has, as expected, efficiency strictly less than one when compared with the normal scores statistic, and the slope of the normal scores test is very close to the maximum possible slope, at normal alternatives.