Moderate deviations for two sample t-statistics

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

Received: 14 July 2006
Revised: 12 December 2006

Abstract

Let X₁,...,X_n1 be a random sample from a population with mean µ₁ and variance , and X₁,...,X_n1 be a random sample from another population with mean µ₂ and variance independent of {X_i,1 ≤ i ≤ n₁}. Consider the two sample t-statistic . This paper shows that ln P(T ≥ x) ~ -x²/2 for any x := x(n₁,n₂) satisfying x → ∞, x = o(n₁ + n₂)^1/2 as n₁,n₂ → ∞ provided 0 < c₁ ≤ n₁/n₂ ≤ c₂ < ∞. If, in addition, E|X₁|³ < ∞, E|Y₁|³ < ∞, then holds uniformly in x ∈ (O,o((n₁ + n₂)^1/6))