Moderate Deviations for Two Sample T-statistics
نویسنده
چکیده
Abstract. Let X1, ..., Xn1 be a random sample from a population with mean μ1 and variance σ 2 1 , and Y1, ..., Yn2 be a random sample from another population with mean μ2 and variance σ 2 2 independent of {Xi, 1 ≤ i ≤ n1}. Consider the two sample t-statistic T = X̄−Ȳ −(μ1−μ2) √ s1/n1+s 2 2/n2 . This paper shows that lnP (T ≥ x) ∼ −x2/2 for any x := x(n1, n2) satisfying x → ∞, x = o(n1 + n2) as n1, n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, in addition, E|X1| < ∞, E|Y1| < ∞, then P (T≥x) 1−Φ(x) → 1 holds uniformly in x ∈ (0, o((n1 + n2))). Mathematics Subject Classification. 60F10, 60G50, 62F05.
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