On smooth change-point location estimation for Poisson Processes

Abstract We are interested in estimating the location of what we call “smooth change-point” from n independent observations an inhomogeneous Poisson process. The smooth change-point is a transition intensity function process one level to another which happens smoothly, but over such small interval, that its length $$\delta _n$$ δ n considered be decreasing 0 as $$n\rightarrow +\infty $$ → + ∞ . show if goes zero slower than 1/ , our model locally asymptotically normal (with rather unusual rate $$\sqrt{\delta _n/n}$$ / ), and maximum likelihood Bayesian estimators consistent, efficient. If, on contrary, faster non-regular behaves like model. More precisely, this case converge at have non-Gaussian limit distributions All these results obtained using ratio analysis method Ibragimov Khasminskii, equally yields convergence polynomial moments estimators. However, order study estimator where cannot applied usual topologies functional spaces. So, should go through use alternative topology will future work.