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2014 When is it no longer possible to estimate a compound Poisson process?
Céline Duval
Electron. J. Statist. 8(1): 274-301 (2014). DOI: 10.1214/14-EJS885

Abstract

We consider centered compound Poisson processes with finite variance, discretely observed over $[0,T]$ and let the sampling rate $\Delta=\Delta_{T}\rightarrow\infty$ as $T\rightarrow\infty$. From the central limit theorem, the law of each increment converges to a Gaussian variable. Then, it should not be possible to estimate more than one parameter at the limit. First, from the study of a parametric example we identify two regimes for $\Delta_{T}$ and we observe how the Fisher information degenerates. Then, we generalize these results to the class of compound Poisson processes. We establish a lower bound showing that consistent estimation is impossible when $\Delta_{T}$ grows faster than $\sqrt{T}$. We also prove an asymptotic equivalence result, from which we identify, for instance, regimes where the increments cannot be distinguished from Gaussian variables.

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Céline Duval. "When is it no longer possible to estimate a compound Poisson process?." Electron. J. Statist. 8 (1) 274 - 301, 2014. https://doi.org/10.1214/14-EJS885

Information

Published: 2014
First available in Project Euclid: 31 March 2014

zbMATH: 1293.62010
MathSciNet: MR3189556
Digital Object Identifier: 10.1214/14-EJS885

Subjects:
Primary: 62B15 , 62K99
Secondary: 62M99

Keywords: compound Poisson process , Discretely observed random process , information loss

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society

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