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Moderate deviationsfor two sample t-statistics

Published online by Cambridge University Press:  19 June 2007

Hongyuan Cao
Hongyuan Cao*
Affiliation:
Department of Statistics and Operations Research, University of North Carolina-Chapel Hill, Chapel Hill, NC 27599, USA; [email protected]
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Abstract

Let X1,...,Xn1 be arandom sample from a population with mean µ 1 and variance $\sigma_1^2$ , and X1,...,Xn1 be a random sample fromanother population with mean µ 2 and variance $\sigma_2^2$ independent of{Xi,1 ≤ i ≤ n1 }.Consider the twosample 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 thatln 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, inaddition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T \geq x)}{1-\Phi(x)} \to 1 $ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

Keywords

Two sample t-statisticasymptotic distributionmoderate deviation.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 264 - 271
Copyright
© EDP Sciences, SMAI, 2007

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References

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