Original scientific paper

A general theorem on approximate maximum likelihood estimation

Miljenko Huzak

Abstract

In this paper a version of the general theorem on approximate maximum likelihood estimation is proved. We assume that there exists a log-likelihood function L(θ) and a sequence (Ln(θ)) of its estimates defined on some statistical structure parametrized by θ from an open set Θ ⊆ Rd, and dominated by a probability P. It is proved that if L(θ) and Ln(θ) are random functions of class C2(Θ) such that there exists a unique point θ ∈ Θ of the global maximum of L(θ) and the first and second derivatives of Ln(θ) with the respect to θ converge to the corresponding derivatives of L(θ) uniformly on compacts in Θ with the order OP(γn), limn γn = 0, then there exists a sequence of Θ-valued random variables θn which converges to θ with the order OP(γn), and such that θn is a stationary point of Ln(θ) in asymptotic sense. Moreover, we prove that under two more assumptions on L and Ln, such estimators could be chosen to be measurable with respect to the σ-algebra generated by Ln(θ).

Keywords

Parameter estimation; consistent estimators; approximate likelihood function

Hrčak ID:

4856

URI

https://hrcak.srce.hr/4856

Publication date:

1.6.2001.

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