Moderate deviations for two sample t-statistics
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 264-271

Let X 1 ,...,X n 1 be a random sample from a population with mean μ 1 and variance σ 1 2 , and Y 1 ,...,Y n 2 be a random sample from another population with mean μ 2 and variance σ 2 2 independent of {X i ,1≤i≤n 1 }.Consider the two sample t-statistic T=X ¯-Y ¯-(μ 1 -μ 2 ) s 1 2 /n 1 +s 2 2 /n 2 . This paper shows that lnP(T≥x)∼-x 2 /2 for any 𝑥:=𝑥(𝑛 1 ,𝑛 2 ) satisfying x→∞, x=o(n 1 +n 2 ) 1/2 as n 1 ,n 2 →∞ provided 0<c 1 ≤n 1 /n 2 ≤c 2 <∞. If, in addition, E|X 1 | 3 <∞, E|Y 1 | 3 <∞, then P(T≥x) 1-Φ(x)→1 holds uniformly in x∈(0,o((n 1 +n 2 ) 1/6 )).

DOI : 10.1051/ps:2007020
Classification : 60F10, 60G50, 62F05
Keywords: two sample t-statistic, asymptotic distribution, moderate deviation 
@article{PS_2007__11__264_0,
     author = {Cao, Hongyuan},
     title = {Moderate deviations for two sample $t$-statistics},
     journal = {ESAIM: Probability and Statistics},
     pages = {264--271},
     year = {2007},
     publisher = {EDP Sciences},
     volume = {11},
     doi = {10.1051/ps:2007020},
     mrnumber = {2320820},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2007020/}
}
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Cao, Hongyuan. Moderate deviations for two sample $t$-statistics. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 264-271. doi: 10.1051/ps:2007020

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