A Generalization of the Beta-Distribution

Abstract

A class of distributions is defined and studied which includes as particular cases (cf. Section 13) the ordinary,1-distribution, the (univariate) triangular distribution, the uniform distribution over any nondegenerate simplex, and a continuous range of other distributions over such a simplex, called basic ,8-distributions (Section 6) and immediately analogous to the ordinary fl-distribu- tion. Our class also includes (Section 13 (vi)) various (univariate and other) distributions which arise in connection with the random division of an interval. The main results are given in Section 2 and further results for the univariate case are given in Section 8. This paper is exclusively concerned with the mathematical theory. One ap- plication may, however, be mentioned, which will be considered in more detail elsewhere. Suppose we wish to test the hypothesis Ho that n - 1 numbers ... Yi, , Yn-1 (all lying between 0 and 1) were drawn independently from a rec- tangular distribution over (0, 1). Let u1, , un be the lengths of the n in- tervals into which the yj divide the interval (0, 1). Then Ho is equivalent to the hypothesis that the point with vector-coordinate u is distributed uniformly over a certain non-degenerate simplex S, and a useful set of alternative hypotheses is the set of basic n-dimenisional d-distributions. Hence (using Section 4) this theory can be used to find the power-functions of certain tests of the hypothe- sis Ho.