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.
“…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 !…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…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).…”
“…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 !…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…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).…”
“…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). …”
Section: ) Is a (Weakly) Consistent Estimator Of λ(S)mentioning
“…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.…”
Abstract. A consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process in the presence of linear trend is constructed and investigated. It is assumed that only a single realization of the Poisson process is observed in a bounded window. We prove that the proposed estimator is consistent when the size of the window indefinitely expands.
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