Limiting distributions of the non-central t-statistic and their applications to the power of t-tests under non-normality

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 T-n = root nX/S, where (X) over bar and S-2 are the sample mean and the sample variance, respectively We refer to T-n as the central or non-central (Student's) t-statistic, depending on whether EX = 0 or EX not equal 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 EX4 = infinity. As an application, we study the effect of non-normality on the performance of the t-test.