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May 2007 Limiting distributions of the non-central $t$-statistic and their applications to the power of $t$-tests under non-normality
Vidmantas Bentkus, Bing-Yi Jing, Qi-Man Shao, Wang Zhou
Bernoulli 13(2): 346-364 (May 2007). DOI: 10.3150/07-BEJ5073


Let $X_1,X_2$,… be a sequence of independent and identically distributed random variables. Let $X$ be an independent copy of $X_1$. Define $\mathbb{T}_{n}=\sqrt{n}\bar{X}/S$, where $\bar{X}$ and $S^2$ are the sample mean and the sample variance, respectively. We refer to $\mathbb{T}_{n}$ as the central or non-central (Student’s) $t$-statistic, depending on whether $\mathrm{E}X=0$ or $\mathrm{E}X≠0$, respectively. The non-central $t$-statistic arises naturally in the calculation of powers for $t$-tests. The central $t$-statistic has been well studied, while there is a very limited literature on the non-central $t$-statistic. In this paper, we attempt to narrow this gap by studying the limiting behaviour of the non-central $t$-statistic, which turns out to be quite complicated. For instance, it is well known that, under finite second-moment conditions, the limiting distributions for the central $t$-statistic are normal while those for the non-central $t$-statistic can be non-normal and can critically depend on whether or not $\mathrm{E}X=∞$. As an application, we study the effect of non-normality on the performance of the $t$-test.


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Vidmantas Bentkus. Bing-Yi Jing. Qi-Man Shao. Wang Zhou. "Limiting distributions of the non-central $t$-statistic and their applications to the power of $t$-tests under non-normality." Bernoulli 13 (2) 346 - 364, May 2007.


Published: May 2007
First available in Project Euclid: 18 May 2007

zbMATH: 1129.60021
MathSciNet: MR2331255
Digital Object Identifier: 10.3150/07-BEJ5073

Keywords: domain of attraction , limit theorems , non-central t-statistic , power of t-test

Rights: Copyright © 2007 Bernoulli Society for Mathematical Statistics and Probability

Vol.13 • No. 2 • May 2007
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