Quasimaximum Likelihood Estimators for Two-parameter Gamma Distributions

Abstract

The Gamma probability distribution is defined by the density function [β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> γ(α)] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> exp (−x/β). This paper presents new estimators for the parameters α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> and β. Required calculations are simple, primarily involving the inner product of certain elementary statistics and their logarithms. Both of the new estimators are shown to be unbiased. The variance formulas for each and their covariance formula are derived—counterparts of these for the method of moments (MM) and the method of maximum likelihood (ML) not being known except in the asymptotic form. Curve-fitting results from samples of size 50 are provided. They support the proposition that the new quasimaximum likelihood estimators (QML) are at least as effective as MM and ML estimators. Illustrative estimation based on samples of size 5 also is presented for comparison; the results highlight relative smaller variances for the ML estimators. The same Monte Carlo results signal a possibility that significant negative bias occurs with both the MM and ML techniques if small samples are used.