Moderate deviations for two sample $t$-statistics

Cao, Hongyuan

ESAIM: Probability and Statistics, Tome 11 (2007), pp. 264-271.

Let $X_1,...,X_{n_1}$ be a random sample from a population with mean ${\mu }_1$ and variance ${\sigma }_1^2$, and $Y_1,...,Y_{n_2}$ be a random sample from another population with mean ${\mu }_2$ and variance ${\sigma }_2^2$ independent of $\{X_i,1\le i\le n_1\}$.Consider the two sample t-statistic $T=\frac{\overline{X}-\overline{Y}-({\mu }_1-{\mu }_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$. This paper shows that $lnP(T\ge x)\sim -x^2/2$ for any $𝑥:=𝑥({𝑛}_{\mathit{1}},{𝑛}_{\mathit{2}})$ satisfying $x\to \infty$, $x=o{(n_1+n_2)}^{1/2}$ as $n_1,n_2\to \infty$ provided $0<c_1\le n_1/n_2\le c_2<\infty .$ If, in addition, $E|X_1{|}^3<\infty$, $E|Y_1{|}^3<\infty$, then $\frac{P(T\ge x)}{1-\Phi \left(x\right)}\to 1$ holds uniformly in $x\in (0,o\left({(n_1+n_2)}^{1/6}\right))$.

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