Estimation of the Location of a 0-type or $\infty$-type Singularity by Poisson Observations

Details Estimation of the Location of a 0-type or $\infty$-type Singularity by Poisson Observations Estimation of the Location of a 0-type or $\infty$-type Singularity by Poisson Observations

2006-11-02

arxiv.org/abs/math/0611043v1

Mathematics

Statistics Theory

Scientific paper

We consider an inhomogeneous Poisson process $X$ on $[0,T]$. The intensity function of $X$ is supposed to be strictly positive and smooth on $[0,T]$ except at the point $\theta$, in which it has either a 0-type singularity (tends to 0 like $\abs{x}^p$, $p\in(0,1)$), or an $\infty$-type singularity (tends to $\infty$ like $\abs{x}^p$, $p\in(-1,0)$). We suppose that we know the shape of the intensity function, but not the location of the singularity. We consider the problem of estimation of this location (shift) parameter $\theta$ based on $n$ observations of the process $X$. We study the Bayesian estimators and, in the case $p>0$, the maximum likelihood estimator. We show that these estimators are consistent, their rate of convergence is $n^{1/(p+1)}$, they have different limit distributions, and the Bayesian estimators are asymptotically efficient.

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