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Statistical Inference for Renewal Processes
Author(s) -
Comte F.,
Duval C.
Publication year - 2018
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12295
Subject(s) - estimator , orthonormal basis , mathematics , upsampling , interval (graph theory) , convolution (computer science) , discretization , projection (relational algebra) , basis (linear algebra) , truncation (statistics) , mathematical optimization , algorithm , computer science , statistics , mathematical analysis , image (mathematics) , physics , quantum mechanics , combinatorics , artificial intelligence , machine learning , artificial neural network , geometry
We consider non‐parametric estimation for interarrival times density of a renewal process. For continuous time observation, a projection estimator in the orthonormal Laguerre basis is built. Nonstandard decompositions lead to bounds on the mean integrated squared error (MISE), from which rates of convergence on Sobolev–Laguerre spaces are deduced, when the length of the observation interval gets large. The more realistic setting of discrete time observation is more difficult to handle. A first strategy consists in neglecting the discretization error. A more precise strategy aims at taking into account the convolution structure of the data. Under a simplifying ‘dead‐zone’ condition, the corresponding MISE is given for any sampling step. 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.