On Approximations to Sampling Distributions of the Mean for Samples From Non-Normal Populations

Abstract

In comparison with the vast number of distribution functions available for describing non-normal populations (see Haight [11]) a very few sampling distributions of the mean, when sampling from these, are known in exact form. The more important results have been derived by Baker [1], [2], Baten [3], [4], Bose [6], Church [7], Hall [12], Irwin [13], [14], Rao [16] and Shrivastava [18]. Except in the case of the well-known result which follows when sampling from a Pearson's Type III population, many of these are of exceptional form and hardly any of them have been tabulated. Furthermore, since most of these results are only of practical use in the case of very small sample sizes, the need for approximations to the sampling distribution of the mean have been felt. In the case of the normal approximation, which has been widely used in many cases in virtue of the Central Limit Theorem, Berry [5] has shown that it is not sufficient for moderate sample sizes. For the latter case, Esseen [9] has obtained approximations in terms of the normal distribution and its derivatives. These results have been extended by Gnedenko and Kolmogorov [10]. Other forms of approximations have been obtained by Daniels [8] and Welker [21]. Another approach to find approximations to fill the gap between the exact sampling distribution and its ultimate normal approximation is presented in this paper. A method developed by Steyn [19] in deriving a differential equation of the moment generating function of the sample mean and variance respectively for samples from a normal population, is used in deriving approximations to sampling distributions of the mean for samples from a number of Pearson's Type populations. Only first order approximations are considered and it will be shown that in the case of sampling from certain skew populations, the results lead to the Pearson's Type III-distribution before approaching normality, whereas, in the symmetrical cases the sampling distribution approaches normality directly.