Maximum Likelihood Estimation of the Parameters of the Beta Distribution from Smallest Order Statistics

Abstract

Numerical methods, useful with high-speed computers, are described for obtaining the maximum likelihood estimat.es of the two (shape) parameters of a beta distribution using the smallest M order statistics, 0 < u 1 ≤ … ≤ ≤ … ≤ u M , in a random sample of size K(≥M). The maximum likelihood estimates are functions only of the ratio, n = M/K, the Mth ordered observation, u M , and the two statistics, G 1 = [II M i=1 u i ]1/M , and G 1 = [II M i=1 (1 – u i )]1/M . For the case of the complete sample (i.e., R = 1), however, the estimates are functions only of G 1 and G 2, and hence, for this case, explicit tables of the estimates are provided. When R < 1, the methods described depend crucially for their usefulness on the availability of a high-speed computer. Some esamples are given of the use of the procedures described for fitting beta distributions to sets of data. In one example, the fit is studied by using beta probability plots.