A Higher Order Approximation to a Percentage Point of the Distribution of a Noncentral t-Statistic Without the Normality Assumption

Abstract

Noncentral distributions appear in two sample problems and are often used in several fields, for example, in biostatistics. A higher order approximation for a percentage point of the noncentral t-distribution under normality is given by Akahira (1995) and is also shown to be numerically better than others. In this article, without the normality assumption, we obtain a higher order approximation to a percentage point of the distribution of a noncentral 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 noncentrality parameter are also given. Further, a numerical comparison of the higher order approximation with the limiting normal distribution is done and the former one 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:

• Chi-statistic
• Confidence limit
• Noncentral t-distribution
• Noncentral t-statistic
• Normal random variable
• percentile

Mathematics Subject Classification:

• Primary 62E17
• Secondary 62F12

Acknowledgements

The authors wish to thank the referee for useful comments.