engleski

A general theorem on approximate maximum likelihood estimation

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 $L(\vartheta )$ and a sequence $(L_n (\vartheta ))$ of its estimates defined on some statistical structure parameterized by $\vartheta$ from an open set $\Theta\subseteq\R^d$, and dominated by a probability $\Pb$. It is proved that if $L(\vartheta )$ and $L_n (\vartheta )$ are random functions of class $C^2 (\Theta )$ such that there exists a unique point $\hat{\vartheta}\in\Theta$ of the global maximum of $L(\vartheta )$ and the first and second derivatives of $L_n = (\vartheta )$ with respect to $\vartheta$ converge to the corresponding derivatives of $L(\vartheta )$ uniformly on compacts in $\Theta$ with the order $O_{\Pb}(\gamma_n )$, $\lim_n\gamma_n =3D0$, then there exists a sequence of $\Theta$-valued random variables $\hat{\vartheta}_n$ which converges to $\hat{\vartheta}$ with the order $O_\Pb (\gamma_n )$ and such that $\hat{\vartheta}_n$ is a stationary point of $L_n (\vartheta )$ in asymptotic sense. Moreover, we prove that under two more assumption on $L$ and $L_n$, such estimators could be chosen to be measurable with respect to the $\sigma$-algebra generated by $L_n (\vartheta )$.

Parameter estimation; consistent estimators; approximate likelihood function.

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