Glasnik Matematicki, Vol. 36, No.1 (2001), 139-153.


Miljenko Huzak

Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia

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 OPn), limn γn = 0, then there exists a sequence of Θ-valued random variables θn which converges to θ with the order OPn), 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).

1991 Mathematics Subject Classification.   62F10, 62F12.

Key words and phrases.   Parameter estimation, consistent estimators, approximate likelihood function.

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