## Electronic Journal of Statistics

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

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

#### 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