This article deals with adaptive nonparametric estimation for Lévy processes observed at low frequency. For general linear functionals of the Lévy measure, we construct kernel estimators, provide upper risk bounds and derive rates of convergence under regularity assumptions. Our focus lies on the adaptive choice of the bandwidth, using model selection techniques. We face here a non-standard problem of model selection with unknown variance. A new approach towards this problem is proposed, which also allows a straightforward generalization to a classical density deconvolution framework.
A Lévy process is observed at time points of distance 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 .Thereby, we encompass the usual low-and high-frequency assumptions and also obtain asymptotics in the mid-frequency regime.
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.
We consider the famous Rasch model, which is applied to psychometric surveys when n persons under test answer m questions. The score is given by a realization of a random binary n × m-matrix. Its (j, k)th component indicates whether or not the answer of the jth person to the kth question is correct. In the mixture Rasch model one assumes that the persons are chosen randomly from a population. We prove that the mixture Rasch model is asymptotically equivalent to a Gaussian observation scheme in Le Cam's sense as n tends to infinity and m is allowed to increase slowly in n. For that purpose we show a general result on strong Gaussian approximation of the sum of independent high-dimensional binary random vectors. As a first application we construct an asymptotic confidence region for the difficulty parameters of the questions.
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