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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:
 6.2e+21 Asymptotic distribution theory in statistics
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