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Asymptotic expansion of the log-likelihood function based on stopping times defined on stochastic processes. (English) Zbl 0585.60054

Mathematical structures, computational mathematics, mathematical modelling 2, Pap. dedic. L. Iliev 70th Anniv., 90-96 (1984).
Summary: [For the entire collection see Zbl 0556.00005.]
Let the parameter space \(\Theta\) be an open subset of \(R^ k\), \(k\geq 1\), and for each \(\theta\in \Theta\), let the r.v.’s \(X_{n'}n\geq 0\), be defined on the probability space (\({\mathcal K}\), \({\mathcal A}\), \(P_{\theta})\) and take values in a Borel subset of a Euclidean space. It is assumed that the finite dimensional distributions of the process \(\{X_{n'}n\geq 0\}\) are of known functional form except that they involve the parameter \(\theta\). For each \(n\geq 1\), let \(v_ n\) be a stopping time defined on this process and have some desirable properties. For \(0<\tau_ n\to \infty\) as \(n\to \infty\), set \(\theta_{\tau_ n}=\theta +h_ n\tau_ n^{-1/2},\) \(h_ n\to h\in R^ k\), and consider the log-likelihood function \(\Lambda v_ n(\theta)\) of the probability measure \(\tilde P_{n,\theta_{\tau_ n}}\) with respect to \(P_{n,\theta}\). Here \(\tilde P_{n,\theta}\) is the restriction of \(P_{\theta}\) to the \(\sigma\)-field induced by the r. v.’s \(X_ 0,...,X_{v_ n}.\)
The main purpose of this paper is to obtain an asymptotic expansion of \(\Lambda v_ n(\theta)\) in the probability sense. The asymptotic distribution of \(\Lambda v_ n(\theta)\), as well as that of another r.v. closely related to it, is also obtained under both \(\tilde P_{n,\theta}\) and \(\tilde P_{n,\theta_{\tau_ n}}\).

MSC:

60G30 Continuity and singularity of induced measures
60F99 Limit theorems in probability theory
60G40 Stopping times; optimal stopping problems; gambling theory

Citations:

Zbl 0556.00005