Pseudo Maximum Likelihood Estimation: Theory and Applications

Let X₁, ⋯, Xₙ be i.i.d. random variables with probability distribution Fθ, p indexed by two real parameters. Let p̂ = p̂(X₁, ⋯, Xₙ) be an estimate of p other than the maximum likelihood estimate, and let θ̂ be the solution of the likelihood equation ∂/∂ θ ln L(x, θ, p̂) = 0 which maximizes the likelihood. We call θ̂ a pseudo maximum likelihood estimate of θ, and give conditions under which θ̂ 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.