# Moderate deviations for two sample t-statistics

ESAIM: Probability and Statistics (2007)

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

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topCao, 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

top- 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
- 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
- V.V. Petrov, Limit Theorems of Probability Theory. Clarendon Press, Oxford (1995). Zbl0826.60001
- J. Robinson, and Q. Wang, On the self-normalized Cramér-type large deviation. J. Theor. Probab.18 (2005) 891–909. Zbl1087.60026
- Q.M. Shao, Self-normalized large deviations. Ann. Probab.25 (1997) 285–328. Zbl0873.60017
- 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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