Nonparametric density estimation in compound Poisson processes using convolution power estimators
Consider a compound Poisson process which is discretely observed with sampling interval <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\Delta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="normal">Δ</mi> </math> </EquationSource> </InlineEquation> until exactly <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>n</mi> </math> </EquationSource> </InlineEquation> nonzero increments are obtained. The jump density and the intensity of the Poisson process are unknown. In this paper, we build and study parametric estimators of appropriate functions of the intensity, and an adaptive nonparametric estimator of the jump size density. The latter estimation method relies on nonparametric estimators of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$m$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>m</mi> </math> </EquationSource> </InlineEquation>th convolution powers density. The <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$L^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi>L</mi> <mn>2</mn> </msup> </math> </EquationSource> </InlineEquation>-risk of the adaptive estimator achieves the optimal rate in the minimax sense over Sobolev balls. Numerical simulation results on various jump densities enlight the good performances of the proposed estimator. Copyright Springer-Verlag Berlin Heidelberg 2014
