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

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 Akahira, M. 1995. A higher order approximation to a percentage point of the non-central t-distribution. Communications in Statistics–Simulation, 24(3): 595–605. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) 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 Akahira, M. 1995. A higher order approximation to a percentage point of the non-central t-distribution. Communications in Statistics–Simulation, 24(3): 595–605. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) 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.