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

Abstract : We consider centered compound Poisson processes with fi nite variance, discretely observed over [0; T] and let the sampling rate $\Delta$ go to infinity as T tends to infinity. 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 and 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 $\Delta$ 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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Submitted on : Monday, August 11, 2014 - 2:03:04 PM
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Céline Duval. When is it no longer possible to estimate a compound Poisson process?. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2014, 8, pp.274-301. ⟨10.1214/14-EJS885⟩. ⟨hal-00877195v2⟩

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