A partially observed poisson process

Let M be a Poisson random measure on [0, [infinity]) and let {X(t): t[epsilon][0,[infinity])} be an alternating renewal process induced by the probability measures [eta] and [mu];i.e., X alternates between the states 1 and 0 with independent sojourns, those in 1 having distribution [eta] and those in 0 having distribution [mu]. Assume that M and X are independent. Let X*M be the random measure defined by X*M(A)=[integral operator]AX(u)M(du). In this paper we study the random measure X*M and its relationship to the processes M and X engendering it. One interpretation is that M is an underlying process of events and X is a screening process that allows or prevents observation of M, so that X*M is the observed process of events. Distributional, structural and asymptotic properties of the random measure X* M are presented. We develop procedures for statistical estimation and filtering for M, based on observations either of both X and X*M or of X* M alone, when M is an ordinary Poisson process. Estimation is of the rate of M; the filtering results are explicit or recursive calculations of conditional expectations of the form E[M(t)|¢t, where (¢s) is an appropriate observed history. We also treat estimation of the mean measure m of M when m is periodic and of the Lévy measure for M when M is additive. Finally, estimation and filtering results are given for X when X is assumed to be Markov; the observed history may be that of X* M alone or that of both M and X* M.