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Local asymptotic mixed normality of log-likelihood based on stopping times. (English) Zbl 0677.62016
Summary: Let \(\Theta\), the parameter space, be an open subset of \(R^ k\), \(k\geq 1\). For each \(\theta\in \Theta\) let the r.v.’s \(X_ m\), \(m=1,2,..\). be defined on the probability space (\({\mathcal X},A,P_{\theta})\) and take values in space (S,\(\zeta)\), where S is a Borel subset of a Euclidean space and \(\zeta\) is the \(\sigma\)-field of Borel subsets of S. It is assumed that the joint probability law of any finite set of such r.v.’s \(\{X_ m\), \(m\geq 1\}\) has some known functional form except the unknown parameter \(\theta\). For \(h\in R^ k\) and a sequence of p.d. matrices \(\delta_ n=\delta_ n^{k\times k}(\theta_ 0)\) set \(\theta^*_ n=\theta^*=\theta_ 0+\delta_ n^{-1}h\), where \(\theta_ 0\) is the true value of \(\theta\), as one value of \(\theta\). For each \(n\geq 1\), let \(\nu_ n\) be stopping time defined on the process, with some desirable properties. Let \(\Lambda_ m(\theta^*,\theta_ 0)\) be the log- likelihood ratio of the probability measure \(P_{m\theta^*}^ w.\)r.t. the probability measure \(P_{m\theta_ 0}\), where \(P_{m\theta}\) is the restriction of \(P_{\theta}\) on \(A_ m=\sigma <X_ 1,X_ 2,...,X_ m>\). Replacing m by \(\nu_ n\) in \(\Lambda_ m(\theta^*,\theta_ 0)\) we get the randomly stopped log-likelihood ratio, namely \(\Lambda_{\nu_ n}(\theta^*,\theta_ 0).\)
The main purpose of this paper is to show that under certain regularity conditions the limiting distribution of \(\Lambda_{\nu_ n}(\theta^*,\theta_ 0)\) is locally asymptotically mixed normal. Two examples are also taken into account.

MSC:
62E20 Asymptotic distribution theory in statistics
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