# Moderate deviations for two sample t-statistics

• Volume: 11, page 264-271
• ISSN: 1292-8100

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## Abstract

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Let X1,...,Xn1 be a random sample from a population with mean µ1 and variance ${\sigma }_{1}^{2}$, and X1,...,Xn1 be a random sample from another population with mean µ2 and variance ${\sigma }_{2}^{2}$ independent of {Xi,1 ≤ i ≤ n1}. Consider the two sample t-statistic $T=\frac{\overline{X}-\overline{Y}-\left({\mu }_{1}-{\mu }_{2}\right)}{\sqrt{{s}_{1}^{2}/{n}_{1}+{s}_{2}^{2}/{n}_{2}}}$. 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 $\frac{P\left(T\ge x\right)}{1-\Phi \left(x\right)}\to 1$ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

## How to cite

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Cao, Hongyuan. "Moderate deviations for two sample t-statistics." ESAIM: Probability and Statistics 11 (2007): 264-271. <http://eudml.org/doc/250113>.

@article{Cao2007,
abstract = { Let X1,...,Xn1 be a random sample from a population with mean µ1 and variance $\sigma_1^2$, and X1,...,Xn1 be a random sample from another population with mean µ2 and variance $\sigma_2^2$ independent of \{Xi,1 ≤ i ≤ n1\}. 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 ≥ 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 $\frac\{P(T \geq x)\}\{1-\Phi(x)\} \to 1$ holds uniformly in x ∈ (O,o((n1 + n2)1/6))},
author = {Cao, Hongyuan},
journal = {ESAIM: Probability and Statistics},
keywords = {Two sample t-statistic; asymptotic distribution; moderate deviation.; two sample t-statistic; moderate deviation},
language = {eng},
month = {6},
pages = {264-271},
publisher = {EDP Sciences},
title = {Moderate deviations for two sample t-statistics},
url = {http://eudml.org/doc/250113},
volume = {11},
year = {2007},
}

TY - JOUR
AU - Cao, Hongyuan
TI - Moderate deviations for two sample t-statistics
JO - ESAIM: Probability and Statistics
DA - 2007/6//
PB - EDP Sciences
VL - 11
SP - 264
EP - 271
AB - Let X1,...,Xn1 be a random sample from a population with mean µ1 and variance $\sigma_1^2$, and X1,...,Xn1 be a random sample from another population with mean µ2 and variance $\sigma_2^2$ independent of {Xi,1 ≤ i ≤ n1}. 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 ≥ 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 $\frac{P(T \geq x)}{1-\Phi(x)} \to 1$ holds uniformly in x ∈ (O,o((n1 + n2)1/6))
LA - eng
KW - Two sample t-statistic; asymptotic distribution; moderate deviation.; two sample t-statistic; moderate deviation
UR - http://eudml.org/doc/250113
ER -

## References

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1. E. Giné, F. Gőtze, and D.M. Mason, When is the student t-statistic asymptotically standard normal? Ann. Probab.25 (1997) 1514–1531.  Zbl0958.60023
2. B.-Y. Jing, Q.M. Shao, and Q.Y. Wang, Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab.31 (2003) 2167–2215.  Zbl1051.60031
3. V.V. Petrov, Limit Theorems of Probability Theory. Clarendon Press, Oxford (1995).  Zbl0826.60001
4. J. Robinson, and Q. Wang, On the self-normalized Cramér-type large deviation. J. Theor. Probab.18 (2005) 891–909.  Zbl1087.60026
5. Q.M. Shao, Self-normalized large deviations. Ann. Probab.25 (1997) 285–328.  Zbl0873.60017
6. Q.M. Shao, A Cramér type large deviation result for student's t-statistic. J. Theor. Probab.12 (1999) 385–398.  Zbl0927.60045

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