Electronic Journal of Statistics

When is it no longer possible to estimate a compound Poisson process?

Céline Duval

Full-text: Open access

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.

Article information

Source
Electron. J. Statist. Volume 8, Number 1 (2014), 274-301.

Dates
First available in Project Euclid: 31 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1396271048

Digital Object Identifier
doi:10.1214/14-EJS885

Mathematical Reviews number (MathSciNet)
MR3189556

Zentralblatt MATH identifier
1293.62010

Subjects
Primary: 62B15: Theory of statistical experiments 62K99: None of the above, but in this section
Secondary: 62M99: None of the above, but in this section

Keywords
Discretely observed random process compound Poisson process information loss

Citation

Duval, Céline. When is it no longer possible to estimate a compound Poisson process?. Electron. J. Statist. 8 (2014), no. 1, 274--301. doi:10.1214/14-EJS885. https://projecteuclid.org/euclid.ejs/1396271048


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