Open Access
August 2014 Asymptotic lower bounds in estimating jumps
Emmanuelle Clément, Sylvain Delattre, Arnaud Gloter
Bernoulli 20(3): 1059-1096 (August 2014). DOI: 10.3150/13-BEJ515

Abstract

We study the problem of the efficient estimation of the jumps for stochastic processes. We assume that the stochastic jump process $(X_{t})_{t\in[0,1]}$ is observed discretely, with a sampling step of size $1/n$. In the spirit of Hajek’s convolution theorem, we show some lower bounds for the estimation error of the sequence of the jumps $(\Delta X_{T_{k}})_{k}$. As an intermediate result, we prove a LAMN property, with rate $\sqrt{n}$, when the marks of the underlying jump component are deterministic. We deduce then a convolution theorem, with an explicit asymptotic minimal variance, in the case where the marks of the jump component are random. To prove that this lower bound is optimal, we show that a threshold estimator of the sequence of jumps $(\Delta X_{T_{k}})_{k}$ based on the discrete observations, reaches the minimal variance of the previous convolution theorem.

Citation

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Emmanuelle Clément. Sylvain Delattre. Arnaud Gloter. "Asymptotic lower bounds in estimating jumps." Bernoulli 20 (3) 1059 - 1096, August 2014. https://doi.org/10.3150/13-BEJ515

Information

Published: August 2014
First available in Project Euclid: 11 June 2014

zbMATH: 06327903
MathSciNet: MR3217438
Digital Object Identifier: 10.3150/13-BEJ515

Keywords: Convolution theorem , Itô process , LAMN property

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability

Vol.20 • No. 3 • August 2014
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