'Change in space’-point estimation, Part I: Lower bound for rates of consistency

Abstract

Given n discrete observations of a homogeneous diffusion process with a piecewise constant diffusion coefficient containing one point of discontinuity p0, we study the semiparametric problem of estimating its 'change in space'- point p_0 in the high-frequency setting. We establish a lower bound for the minimax rate of convergence n^--3/4, which is slower than the n^-1-rate in traditional change-point problems.