Pseudo Maximum Likelihood Estimation: Theory and Applications

1981/07/01 by Gail Gong, Francisco J. Samaniego · 408 citations
Environmental Science · Mathematics · #Advanced Statistical Methods and Models #Applied mathematics #Combinatorics #Convolution (computer science) #Estimation theory #Estimator #Likelihood function #Likelihood principle #Mathematics #Maximum likelihood #Maximum likelihood sequence estimation #Quasi-maximum likelihood #Restricted maximum likelihood #Soil Geostatistics and Mapping #Statistical Methods and Inference #Statistics

paper · doi:10.1214/aos/1176345526

published in The Annals of Statistics 9(4) (Institute of Mathematical Statistics)

openalex publication_date 1981/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

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

Citations

• On Stochastic Limit and Order Relationships

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