Pseudo Maximum Likelihood Estimation: Theory and Applications

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. random variables with probability distribution $F_{\theta, p}$ indexed by two real parameters. Let $\hat{p} = \hat{p}(X_1, \cdots, X_n)$ be an estimate of $p$ other than the maximum likelihood estimate, and let $\hat{\theta}$ be the solution of the likelihood equation $\partial/\partial \theta \ln L(\mathbf{x}, \theta, \hat{p}) = 0$ which maximizes the likelihood. We call $\hat{\theta}$ a pseudo maximum likelihood estimate of $\theta$, and give conditions under which $\hat{\theta}$ 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 convolution, models which exhibit such behavior and satisfy the regularity conditions 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.