A Higher Order Approximation to the Distribution of a Non-Central t-Statistic under Non-Normality
Nao Ohyauchi1, Masafumi Akahira2, Shinichi Kawai3
1Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, Japan; 2Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, Japan; 3National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Ibaraki, Japan

In a usual case, the distribution of the statistic based on a random sample from the normal distribution with mean 0, under non-zero mean, is called the non-central distribution. the non-central distribution is useful for calculating the power in hypothesis testing, obtaining the confidence interval of a non-centrality parameter. In such a case a percentage point of the non-central distribution plays an important part, but it is not easy to get the percentage point analytically. So, a higher order approximation for a percentage point of the non-central t-distribution under normality is given by Akahira (1995, Commun. Statist. - Simula., 24, pp. 595-605) and is also shown to be numerically better than others. In this article, without the normality assumption, we obtain a higher order approximation to the percentage point of the distribution of a non-central t-statistic, in a similar way to Akahira (1995) where the statistic based on a linear combination of a normal random variable and a chi-statistic takes an important role. Its application to the confidence limit and the confidence interval for a non-centrality parameter are also given. Further, a numerical comparison of the higher order approximation with the limiting normal distribution is done and the former is shown to be more accurate. As a result of the numerical calculation, the higher order approximation seems to be useful in practical situations, when the size of sample is not so small.

Keywords: Percentage point; Normal random variable; Chi-statistic; Confidence limit

Biography: I got the degree of D. Sci., mathematics, from University of Tsukuba in 2004, and was a researcher and a research associate, University of Tsukuba from 2004 to 2007.

I have been an assistant professor, University of Tsukuba since 2007. My research interest is on statistical estimation and its related topics.