Large deviation probabilities in estimation of Poisson random measures
The paper deals with a homogeneous Poisson random measure \(N\) on \(R_+\times E\), defined on a filtered probability space, where \(E\) is a Polish space. The intensity measure \(q\) of \(N\) is given by \(q(dt, dx) = \nu (dx)dt\), where \(\nu \) is a \(\sigma \)-finite measure on \(E\). The problem is to estimate the Lévy measure \(\nu \) when \(N\) is observed over a fixed time interval \([0,T]\). The authors investigate the exponential rate of parametric estimators of \(\nu \) with regard to the probabilities of large deviations. Following the approach of \textit{R. R. Bahadur, S. L. Zabell} and \textit{J. C. Gupta} [in: Asymptotic theory of statistical tests and estimation, 33-64 (1980; Zbl 0601.62037)], they state a lower bound for an asymptotic \(T\rightarrow \infty \). The main goal of this paper is to provide an upper bound and a large deviation principle (LDP) for a maximum likelihood estimator (MLE). Since in general a MLE cannot be explicitly expressed in function of \(N\) but only as an implicit solution of the likelihood equation, an implicit contraction principle is stated and applied to derive from a LDP for \(N\), a LDP for a MLE, and to identify the rate function.
scientific article; zbMATH DE number 1364490
scientific article