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Efficient nonparametric inference for discretely observed compound Poisson processes

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Abstract

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.

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Notes

  1. In personal communication with M. Trabs, it was confirmed that the function \({{\mathrm{\mathbbm {1}}}}_{(-\infty ,t]}\) in Corollary 4.5 in [35] should be \(f^{(N)}_t\) for their results to be correct, and \(\tilde{\chi }_\nu \), defined after the corollary, should equal \(\lambda f_t^{(F)}\). We also confirmed the covariance of the limiting process in the general case considered in [5] does not coincide with \(\varSigma _{s,t}^{F}\).

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Authors and Affiliations

  1. Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB30WB, UK

    Alberto J. Coca

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  1. Alberto J. Coca
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Correspondence to Alberto J. Coca.

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The author is grateful to Richard Nickl for the numerous discussions leading to the completion of this work, to the anonymous referees for their insightful remarks and to Fundación “La Caixa”, EPSRC (Grant EP/H023348/1 for the Cambridge Centre for Analysis) and Fundación Mutua Madrileña for their generous support.

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Coca, A.J. Efficient nonparametric inference for discretely observed compound Poisson processes. Probab. Theory Relat. Fields 170, 475–523 (2018). https://doi.org/10.1007/s00440-017-0761-5

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