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Efficient nonparametric inference for discretely observed compound Poisson processes. (English) Zbl 1383.62081
Summary: A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as confidence regions and tests are briefly discussed.

MSC:
62G05 Nonparametric estimation
60G51 Processes with independent increments; Lévy processes
60F05 Central limit and other weak theorems
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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