Pseudo Maximum Likelihood Estimation: Theory and Applications

Abstract

Let Xi, * * *, X, be i.i.d. random variables with probability distribution F9,p indexed by two real parameters. Let p = p (XI, * * *, X ) be an estimate of p other than the maximum likelihood estimate, and let 6 be the solution of the likelihood equation aaSO ln L(x, 0, 6) = 0 which maximizes the likelihood. We call # a pseudo maximum likelihood estimate of 0, and give conditions under which # is consistent and asymptotically normal. Pseudo maximum likelihood estimation easily extends to k-parameter models, and is of interest in problems in which the likelihood surface is ill-behaved in higher dimensions but well- behaved in lower dimensions. We examine several signal-plus-noise, or con- volution, models which exhibit such behavior and satisfy the regularity con- ditions of the asymptotic theory. For specific models, a numerical comparison of asymptotic variances suggests that a pseudo maximum likelihood estimate of the signal parameter is uniformly more efficient than estimators proposed previously