Nonparametric adaptive estimation of linear functionals for low frequency observed Lévy processes

For a Lévy process X having finite variation on compact sets and finite first moments, µ( dx) = xv( dx) is a finite signed measure which completely describes the jump dynamics. We construct kernel estimators for linear functionals of µ and provide rates of convergence under regularity assumptions. Moreover, we consider adaptive estimation via model selection and propose a new strategy for the data driven choice of the smoothing parameter.Statistics of stochastic processes, Low frequency observed Lévy processes, Nonparametric statistics, Adaptive estimation, Model selection with unknown variance

Density deconvolution from repeated measurements without symmetry assumption on the errors

We consider deconvolution from repeated observations with unknown error distribution. So far, this model has mostly been studied under the additional assumption that the errors are symmetric. We construct an estimator for the non-symmetric error case and study its theoretical properties and practical performance. It is interesting to note that we can improve substantially upon the rates of convergence which have so far been presented in the literature and, at the same time, dispose of most of the extremely restrictive assumptions which have been imposed so far

Estimation of the characteristics of a Lévy process observed at arbitrary frequency

A Lévy process is observed at time points of distance Δ until time T. We construct an estimator of the Lévy-Khinchine characteristics of the process and derive optimal rates of convergence simultaneously in T and Δ. Thereby, we encompass the usual low- and high-frequency assumptions and obtain also asymptotics in the mid-frequency regime.Jump process, Lévy measure, deconvolution problem, statistical inverse problem

Estimation of the characteristics of a Lévy process observed at arbitrary frequency

A Lévy process is observed at time points of distance delta until time T. We construct an estimator of the Lévy-Khinchine characteristics of the process and derive optimal rates of convergence simultaneously in T and delta. Thereby, we encompass the usual low- and high-frequency assumptions and obtain also asymptotics in the mid-frequency regime.Lévy process, Lévy-Khinchine characteristics, Nonparametric estimation, Inverse problem, Optimal rates of convergence