Statistical inference for renewal processes

Abstract : We consider nonparametric density estimation for interarrival times density of a renewal process. If it is possible to get continuous observation of the process, then a projection estimator in an orthonormal functional basis can be built; we choose to work on R+ with the Laguerre basis. Nonstandard decompositions can lead to bounds on the mean integrated squared error (MISE), from which rates of convergence on Sobolev-Laguerre spaces can be deduced, when the length of the observation interval gets large. The more realistic setting of discrete time observation with sampling rate $\Delta$ is more difficult to handle. A first strategy consists in neglecting the discretization error, and under suitable conditions on $\Delta$, an analogous MISE bound is obtained. A more precise strategy aims at 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 $\Delta$ as well as for small $\Delta$. 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.
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Fabienne Comte, Céline Duval. Statistical inference for renewal processes. Scandinavian Journal of Statistics, Wiley, 2018, ⟨10.1111/sjos.12295⟩. ⟨hal-01349443v3⟩

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