Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme
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A compound Poisson process whose jump measure and intensity are unknown is observed at finitely many equispaced times and a purely data-driven wavelet-type estimator of the Lévy density ν is constructed through the spectral approach. Assuming minimal tail assumptions, it is shown to estimate ν at the best possible rate of convergence over Besov balls under the losses L^p(R), p∈[1,∞], and robustly to the observation regime (high- and low-frequency). The adaptive estimator is obtained by applying Lepskiĭ's method and, thus, novel exponential-concentration inequalities are proved including one for the uniform fluctuations of the empirical characteristic function. These are of independent interest, as are the proof-strategies employed to depart from the ubiquitous quadratic structure and to show robustness to the observation scheme without polynomial-moment conditions.
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