ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, June 2007, Vol. 11, pp. 264-271
DOI: 10.1051/ps:2007020

Moderate deviations for two sample t-statistics

Hongyuan Cao

Department of Statistics and Operations Research, University of North Carolina-Chapel Hill, Chapel Hill, NC 27599, USA; hycao@email.unc.edu

(Received July 14, 2006. Revised December 12, 2006. Published online 19 June 2007.)

Abstract
Let X₁, ..., X_n₁ be a random sample from a population with mean $\mu_1$ and variance $\sigma_1^2$ , and Y₁, ..., Y_n₂ be a random sample from another population with mean $\mu_2$ and variance $\sigma_2^2$ independent of $\{X_i, 1 \leq i \leq n_1\}$ . Consider the two sample t-statistic $T={{\bar X-\bar Y-(\mu_1-\mu_2)} \over \sqrt{s_1^2/n_1+s_2^2/n_2}}$ . This paper shows that $\ln P(T \geq x) \sim -x^2/2 \$ for any ${\it x := x(n_1,n_2)}$ satisfying $x \to \infty$ , x = o(n₁+n₂)^1/2 as $n_1, n_2 \to \infty$ provided $0 < c_1 \leq n_1/n_2 \leq c_2 < \infty.$ If, in addition, $E\vert X_1\vert^3 < \infty$ , $E\vert Y_1\vert^3 < \infty$ , then $\frac{P(T \geq x)}{1-\Phi(x)} \to 1$ holds uniformly in $x \in(0,o((n_1+n_2)^))$ .

Mathematics Subject Classification. 60F10, 60G50, 62F05

Key words: Two sample t-statistic, asymptotic distribution, moderate deviation.

© EDP Sciences, SMAI 2007