Likelihood inference for mixtures: Geometrical and other constructions of monotone step-length algorithms

Abstract

This paper considers algorithms for the maximum likelihood estimator of the mixing distribution of a family of parametric densities. Specific interest is devoted to the monotonicity of the step-length of the solution. In ξ 2 we introduce the idea of estimating the area above the second derivative curve, and in Theorem 1 we relate area-overestimation to the monotonicity of the associated algorithm. In ξ 3 we apply Theorem 1 to the monotonicity analysis of well-known algorithms as well as to the construction of a new class of monotone step-length choices which is particularly simple in the mixture setting. Numerical comparisons and refinements in keeping the monotonic step-length from becoming too conservative are given. Theorem 3 characterizes conditions under which the Newton-Raphson step is monotonic and, if these conditions fail to hold, states that the regula falsi step is monotonic.