Risk bounds for the non-parametric estimation of Lévy processes

Abstract

Estimation methods for the Lévy density of a\nLévy process are developed under mild qualitative\nassumptions.\nA classical model selection approach made up\nof two steps is studied. The first step consists\nin the selection of a good estimator, from an approximating\n(finite-dimensional) linear model $\\calS$ for the true\nLévy density.\nThe second is a data-driven selection of a\nlinear model $\\calS$, among a given collection\n$\\{\\calS_{m}\\}_{m\\in\\calM}$,\nthat approximately realizes the best trade-off between the\nerror of estimation within $\\calS$\nand the error incurred when approximating the\ntrue Lévy density by the linear model $\\calS$.\nUsing recent concentration inequalities for functionals\nof Poisson integrals, a bound for the risk of estimation\nis obtained. As a byproduct,\noracle inequalities and long-run\nasymptotics for spline estimators are derived.\nEven though\nthe resulting underlying statistics are based on continuous\ntime observations of the process, approximations\nbased on high-frequency discrete-data can be easily devised.