A stochastic approximation algorithm for maximum‐likelihood estimation with incomplete data

Abstract

AbstractWe propose a new stochastic approximation (SA) algorithm for maximum‐likelihood estimation (MLE) in the incomplete‐data setting. This algorithm is most useful for problems when the EM algorithm is not possible due to an intractable E‐step or M‐step. Compared to other algorithm that have been proposed for intractable EM problems, such as the MCEM algorithm of Wei and Tanner (1990), our proposed algorithm appears more generally applicable and efficient. The approach we adopt is inspired by the Robbins‐Monro (1951) stochastic approximation procedure, and we show that the proposed algorithm can be used to solve some of the long‐standing problems in computing an MLE with incomplete data. We prove that in general O(n) simulation steps are required in computing the MLE with the SA algorithm and O(n log n) simulation steps are required in computing the MLE using the MCEM and/or the MCNR algorithm, where n is the sample size of the observations. Examples include computing the MLE in the nonlinear error‐in‐variable model and nonlinear regression model with random effects.