Statistical properties of a kernel-type estimator of the intensity function of a cyclic Poisson process

Abstract

We consider a kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process when the period is unknown. We assume that only a single realization of the Poisson process is observed in a bounded window which expands in time. We compute the asymptotic bias, variance, and the mean squared error of the estimator when the window indefinitely expands.

Predicting a cyclic Poisson process

“…which is the kernel-type estimator of the intensity function¸of X introduced in Helmers et al (2003) and investigated also in Helmers et al (2005). Here, h n is a sequence of positive real numbers such that h n # 0, as n !…”

“…This assumption is a rather mild one since the set of all Lebesgue point of¸is dense in R, whenever¸is assumed to be locally integrable. The Lebesgue point assumption also occurs in Helmers et al (2003) and Helmers et al (2005).…”

A non-parametric estimator for the doubly periodic Poisson intensity function

“…If all the amplitudes are equal to 1, then the harmonic mean is equal to 1 and thus statement (3.7) reduces to a special case of statement (3.4) in Helmers, Mangku and Zitikis (2005). …”

Consistency of a Kernel-Type Estimator of the Intensity of the Cyclic Poisson Process With the Linear Trend

“…a = 0, (cf. [3], [4], [6], section 2.3 of [7]) to the more general model (1). Suppose now that, for some ω ∈ Ω, a single realization N (ω) of the Poisson process N defined on a probability space (Ω, F, P) with intensity function λ (cf.…”