Estimation of the location of a 0-type or ∞-type singularity by Poisson observations

Author： Dachian Sergueï  

Publisher： Taylor & Francis Ltd

ISSN： 0233-1888

Source： Statistics, Vol.45, Iss.5, 2011-10, pp. : 509-523

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Abstract

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 θ, in which it has either a 0-type singularity (tends to 0 like |X| p , p∈(0, 1)), or an ∞-type singularity (tends to ∞ like |X| p , p∈(−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 θ 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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