Asymptotic effiency of statistical estimates in compound Poisson model. (Ukrainian, English) Zbl 1053.62094

Kukush, O. G.; Mishura, Yu. S.

Teor. Jmovirn. Mat. Stat. 68, 61-73 (2003); translation in Theory Probab. Math. Stat. 68, 67-80 (2004).

The authors deal with estimation of parameters of a stochastic process \(X(t)=\sum_{k=0}^{N(t)}(1+\xi_{k})\), where \(N(t)\) is a homogeneous Poisson process with parameter \(\lambda_{p}>0\), \(\xi_{k}\) has a Poisson distribution with parameter \(\lambda_{D}>0\), \(N(t)\) and \(\xi_{k},\;k\geq1\) are independent. Let \(\nu(t,A)\) be the number of jumps of the process \(X(t)\) till the moment \(t\), with values of jumps from the set \(A\subset {\mathbb N}\). The authors obtain the maximum likelihood estimates of \(\lambda_{p}\) and \(\lambda_{D}\): \[ \widehat\lambda_{p}= \nu(t,[1,\infty))/ t,\quad \widehat\lambda_{d}= \nu^{-1} (t,[1,\infty)) \sum_{k=2}^{\infty}(k-1)\nu(t,\{ k\}), \] and prove that these estimates are consistent, locally asymptotic normal and asymptotically efficient.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

62M09	Non-Markovian processes: estimation
62F12	Asymptotic properties of parametric estimators
62M05	Markov processes: estimation; hidden Markov models
62F10	Point estimation

Keywords:

asymptotic effiency; statistical estimates; compound Poisson model; maximum likelihood estimates