Bernoulli

  • Bernoulli
  • Volume 20, Number 3 (2014), 1059-1096.

Asymptotic lower bounds in estimating jumps

Emmanuelle Clément, Sylvain Delattre, and Arnaud Gloter

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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.

Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1059-1096.

Dates
First available in Project Euclid: 11 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1402488934

Digital Object Identifier
doi:10.3150/13-BEJ515

Mathematical Reviews number (MathSciNet)
MR3217438

Zentralblatt MATH identifier
06327903

Keywords
convolution theorem Itô process LAMN property

Citation

Clément, Emmanuelle; Delattre, Sylvain; Gloter, Arnaud. Asymptotic lower bounds in estimating jumps. Bernoulli 20 (2014), no. 3, 1059--1096. doi:10.3150/13-BEJ515. https://projecteuclid.org/euclid.bj/1402488934


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