Some problems arising in approximating to probability distributions, using moments

Abstract

In the history of the development of statistical distribution theory there have been many instances where it has been possible to determine the sampling moments of the distribution of a statistic, without any immediate prospect of deriving the mathematical distribution itself in explicit form. Two examples of this are the distributions of (i) the Cram6r-von Mises-Simirnoff statistic WN (or NW2) I and (ii) the standardized fourth moment b2 = M4/in2 in samples from a normal universe (where m8 is the sth central sample moment). In so far as there may be a number of alternative mathematical forms which could be used to approximate the unknown true distribution, the question arises as to how to select between them. Suppose, for example, that we take two different frequency functions each having the same first four moments as the unknown true distribution, should we expect that the empirical function whose higher moments are the closer to the true values will give the better representation? And what do we mean by better representation? In so far as a distribution can be represented by a Gram-Charlier or Fisher-Cornish type of expansion we might expect in theory that agreement in moments would lead to agreement in probability integrals, but it is well known that questions of the convergence of such expansions arise in the case of distributions which are far from normal. The distributions (i) and (ii) referred to above are indeed extremely leptokurtic. In the following paper it is proposed to draw together several hitherto unpublished investigations, some dating back a number of years, which bear on these points. In particular, we shall: (a) Consider the proportionate contributions, arising from different parts of the parent frequency, to each of the first six moments of certain selected distributions. (b) Make a comparison of the distribution functions of three leptokurtic distributions, namely (i) the Pearson type IV, (ii) the non-central t, and (iii) Johnson's Su, when their first four moments have identical values. (c) Apply some of the conclusions drawn from the studies (a) and (b) to the problem of determining significance points for the moment ratio statistics Vb1 = m3/ml and b2 m 2 used in testing for departure from normality. To help the comparison of distributions, we shall represent them as points on a (f1,82) chart as illustrated in Figs. 2 and 3, where