Minimal Variance and its Relation to Efficient Moment Tests

Abstract

When a curve is fitted to a set of data by moments, the usual procedure used in testing the hypothesis that the population is of the given form with the parameters as computed from the moments is to compare the higher moments with their expected values as determined by the hypothesis. Generally speaking, moments about the mean are computed although the reason for this is not clear. To shed some light on this question, the sample given in the introduction is fitted to two curves. Moments about various points are compared with their expected values and the discrepancy in standard units examined. This discrepancy is found to vary widely and to have a maximum. The notion of equivalent moment tests is introduced, and on this basis the most efficient moment test is defined in such a way that of all equivalent moment tests, this one is most likely to reject a false hypothesis. For any moment it is shown that there is a point about which its variance is a minimum. The conditions are found which determine the position of this point for second and third moments. It is proved that for symmetrical populations the variance is minimal when the moments are computed about the mean of the population. If the population is an asymmetrical Pearson frequency function, it is proved that the point about which the third moment variance is minimal differs more from the mean that does the corresponding point for second moments. The condition is pointed out for which this is true in the general case. The third and fourth standard semi-invariants of second moments of minimal variance are computed and compared to those of the second moment about the mean. The ratios of these are displayed for some populations to illustrate how this may be used to investigate when the approach to normality is more rapid in one case than in the other. Some examples are presented to contrast these and other tests.