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This paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the -risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator....
Given a sample from a discretely observed Lévy process X = (Xt)t≥0 of the finite jump activity, the problem of nonparametric estimation of the Lévy density ρ corresponding to the process X is studied. An estimator of ρ is proposed that is based on a suitable inversion of the Lévy–Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of ρ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.
This paper deals with order identification for Markov chains with Markov
regime (MCMR) in the context of finite alphabets. We define the joint order
of a MCMR process in terms of the number k of states of the hidden Markov
chain and the memory m of the conditional Markov chain. We study the
properties of penalized maximum likelihood estimators for the unknown order
(k, m) of an observed MCMR process, relying on information theoretic
arguments. The novelty of our work relies in the joint...
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