2006/11/02 by Serguei Dachian, Dachian, Serguei
Mathematics · #62M05 #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:62M05 #stat.TH
paper · pdf · doi:10.48550/arxiv.math/0611043
arxiv created 2006/11/02 · arxiv updated 2009/12/01
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 \absxp, p∈(0,1)), or an ∞-type singularity (tends to ∞ like \absxp, 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 n1/(p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.