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Title: On smooth change-point estimation for Poisson processes Authors:  Serguei Dachian - Universite de Lille (France) [presenting]
Arij Amiri - Universite de Lille (France)
Abstract: Suppose $n$ independent realizations of an inhomogeneous Poisson process are observed whose intensity function goes from one given level to another in a quick (but smooth) manner in the vicinity of an unknown point $\theta$. The size $\delta_n$ of the vicinity is supposed to converge to zero as $n$ goes to infinity. It turns out that the behavior of the maximum likelihood and Bayesian estimators (MLE and BEs) of $\theta$ strongly depends on the rate of convergence of $\delta_n$ to zero. We show that if this convergence is slow, the problem remains regular and the MLE and BEs are asymptotically normal (with a rate comprised between $1/\sqrt{n}$ and $1/n$) and asymptotically efficient. While if the convergence is fast, we show that --- like in the discontinuous case --- the rate of convergence of the estimators is $1/n$, their limiting laws are no longer Gaussian, and only BEs are asymptotically efficient.