## Bernoulli

- Bernoulli
- Volume 13, Number 2 (2007), 346-364.

### 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, and Wang Zhou

#### Abstract

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.

#### Article information

**Source**

Bernoulli, Volume 13, Number 2 (2007), 346-364.

**Dates**

First available in Project Euclid: 18 May 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1179498752

**Digital Object Identifier**

doi:10.3150/07-BEJ5073

**Mathematical Reviews number (MathSciNet)**

MR2331255

**Zentralblatt MATH identifier**

1129.60021

**Keywords**

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

#### Citation

Bentkus, Vidmantas; Jing, Bing-Yi; Shao, Qi-Man; Zhou, Wang. Limiting distributions of the non-central $t$-statistic and their applications to the power of $t$-tests under non-normality. Bernoulli 13 (2007), no. 2, 346--364. doi:10.3150/07-BEJ5073. https://projecteuclid.org/euclid.bj/1179498752