Discretely Observing a White Noise Change-Point Model in the Presence of Blur

Abstract

In discretely observed diffusion models, inference about unknown parameters in a smooth drift function has attracted much interest of late. This paper deals with a diffusion-type change-point model where the drift has a discontinuity across the point of change, analysed in detail in continuous time by Ibragimov and Hasminskii. We consider discrete versions of this model with integrated or blurred observations at a regular lattice. Asymptotic convergence rates and limiting distributions are given for the maximum likelihood change-point estimator when the observation noise and the lattice spacing simultaneously decrease. In particular, it is shown that the continuous and discrete model convergence rates are generally equal only up to a constant; under specific conditions on the blurring function this constant equals unity, and the normalized difference between the estimators tends to zero in the limit.