Nonparametric estimation of the interarrival distribution of a renewal process

Fabienne Comte (1) , Céline Duval (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

In this paper, we consider nonparametric density estimation for interarrival times density of a renewal process. First, we assume continuous observation of the process and build a projection estimator in the Laguerre basis. We study its mean integrated squared error (MISE) and compute rates of convergence on Sobolev-Laguerre spaces when the length of the observation interval gets large. Second, we consider a discrete time observation with sampling rate ∆. A first strategy consists in neglecting the discretization error, and under suitable conditions on ∆, an analogous MISE is obtained. Then, taking into account the structure of the data, a deconvolution estimator is defined and studied. In that case, we work under a simplifying " dead-zone " condition. The MISE corresponding to this strategy is given for fixed ∆ as well as for small ∆. In the three cases, an automatic model selection procedure is described and gives the best MISE, up to a logarithmic term. The results are illustrated through a simulation study.

Domaines: Statistiques [math.ST]

Fabienne Comte, Céline Duval. Nonparametric estimation of the interarrival distribution of a renewal process. 2016. ⟨hal-01349443v1⟩