The Regions of Unimodality and Positivity in the Abbreviated Edgeworth and Gram-Charlier Series

原文摘要

0. On the notation. In the literature it is common to use , and oto denote, respectively, the population mean and standard deviation and so these are used here. Since there is more than one symbol prevalent these days for the coefficient of skewness it is well to state at the outset that in this paper a! is used to denote this parameter. In practice, these population parameters are usually unknown and estimated from the sample. 1. Introductory remarks. The methods of fitting a curve to an observed frequency distribution when the mathematical form of the frequency function is unknown are considerable in number. Many of these methods are discussed at length in [1, 2, 3, and 4]. The statistician faced with the problem of fitting a curve without much a priori knowledge is familiar with at least some of these various methods and with the fact that each method has its disadvantages and restrictions. Occasionally, it is appropriate and convenient to use the Edgeworth or Gram-Charlier series (see, e.g., [1]) as an approximate theoretical model in computing ordinates and thus estimating frequencies of an unknown frequency distribution. As is well known, the use of these series involves the population moments or, more often, estimates of the population moments obtained from the sample. In practical application, it is usual to make use of the first four moments, from which the mean, variance, skewness, and kurtosis, or their estimates are obtained, and apply these in the first three terms of the series in developing the ordinates of a frequency curve. But because the Edgeworth and Gram-Charlier curves are multimodal and/or produce negative values of the ordinate under certain conditions of skewness and kurtosis, their use is not as general as one might prefer. Barton and Dennis [5] have recently derived the conditions on the skewness and kurtosis under which these two curves are multimodal and positive definite. It happens not infrequently, however, that in practical application an estimate of the fourth moment (and therefore the kurtosis) is not available, or for reasons of economy or expediency is not calculated, and it is desired then to use only the first two terms of the Edgeworth series (which are also the first two terms of the Gram-Charlier series), herein referred to as the abbreviated series, in estimating the ordinates or theoretical frequencies from the data of the sample. One can also envisage circumstances in which the statistician contemplating the use of these series has reason to believe that his data are reasonably mesokurtic in distribution and that his chief problem is that of correcting for skewness. In this event again, it might be concluded that the abbreviated series is, for practical purposes, as useful as the series would be if more terms were retained.'

摘要来源：OpenAlex 索引摘要（稿件版本未注明）