CONSISTENCY OF KERNEL-TYPE ESTIMATORS FOR THE FIRST AND SECOND DERIVATIVES OF A PERIODIC POISSON INTENSITY FUNCTION

Abstract

We construct and investigate consistent kernel-type estimators for the first and second derivatives of a periodic Poisson intensity function when the period is known. We do not assume any particular parametric form for the intensity function. More- over, we consider the situation when only a single realization of the Poisson process is available, and only observed in a bounded interval. We prove that the proposed estimators are consistent when the length of the interval goes to infinity. We also prove that the mean-squared error of the estimators converge to zero when the length of the interval goes to infinity.
1991 Mathematics Subject Classification: 60G55, 62G05, 62G20.