Cybernetics and Systems Analysis, Vol. 32, No. 1. 1996 ASYMPTOTIC PROPERTIES OF A NONPARAMETRIC INTENSITY ESTIMATOR OF A NONHOMOGENEOUS POISSON PROCESS A. Ya. Dorogovtsev and A. G. Kukush UDC 519.21 The interest in parameter estimation of Poisson processes [1, 2] is attributable to the numerous applications of these processes in engineering, as well as in physics, biology, and geology. One of the typical cases from the point of view of the mathematical formulation of the problem is described in [1]. When a signal {>,(t): 0 < t < T} is transmitted by modulation of intensity h of a single-mode laser, the photoelectron multiplier at the receiving end picks up the emission electrons. Let/f/(t) be the number of electrons counted up to time t. Then the probabilistic properties of the process {/V(t): 0 < t _< T} are close to those of the Poisson process {N(t): 0 _< t _< T} with intensity function s(t) = h(t) + ho, t E [0, T], where ~x o is the intensity of homogeneous Poisson noise. In some cases the function h is periodic with a known period r > 0. Additional information may be available about the function h, i.e., its membership in a certain class of functions. Below we consider the estimation of the periodic function X from a certain class of functions using observations of the process {At(t): 0 < t < T}. The distribution of the Poisson process is known, and we naturally consider estimators that are best in a certain sense, specifically the maximum likelihood estimators (MLE). We give the conditions on the class of functions that ensure closeness of the estimator to the unknown function for large T, i.e., produce reliable transmission of the signal X in the above-described problem. We also investigate other properties of the estimator. The application of these estimators in observations of the stationary process at the points of a nonhomogeneous Poisson process is briefly discussed in the concluding section. The paper utilizes the procedure for the analysis of the infinite- dimensional parameter estimator in nonlinear regression (see [7, 3, 4]). All the spaces in our study are real. 1. STATEMENT OF THE PROBLEM AND THE ESTIMATOR Let X be the space of right-continuous piecewise-constant functions x: [0, + oo) --, R, such that x(0) = 0 and the magnitude of the discontinuity at each discontinuity point equals 1. The space of functions with the same properties but .with the domain of definition [0, T] is denoted by XT; B r is the a-algebra generated by cylindrical sets in X T. Also let (f~, :3~, P) be a complete probability space. All the stochastic processes considered below are detrmed on this space. Definition 1. Let h: [(3, + ~) --, [0, + oo) be a Lebesgue-measurable function integrable on every finite interval. The process {N(t): t >_ 0} is called a Poisson process with intensity 3, if it has independent increments and for each t 1, t2, 0 < t 1 q < t2, the increment N(t2) - N(tl) is Poisson-distributed with the parameter f2(Odt. t I We know that the paths of the Poisson process {N(t): t > 0} are contained in X. Note that the function a(t) : = t f X(u)du, t > O, is a compensator of the point process from Definition 1. 0 We use the following assumptions. (i) For T > 0, {N(t) : t E [0, T]} is a Poisson process with a fixed but unknown intensity X 0. (ii) The function ~ is periodic with a known period r, r < T. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 91-104, January-February, 1996. Original article submitted July 13, 1994. 74 1060-0396/96/3201-0074515.00 o1996 Plenum Publishing Corporation