Estimation of the convergence rate for the distributions of normalized maximum likelihood estimators in the case of a discontinuous density (Q1358025)

Let \(X_1,\dots, X_n\) be an i.i.d. sample with a p.d.f. \(f(x,\theta)\), \(\theta\in(a,b)\subset \mathbb{R}\). For each \(\theta\), there is a unique discontinuity \(x(\theta)\), with \[ 0\neq q(\theta)= f(x(\theta)-0,\theta)\neq f(x(\theta)+0,\theta)= p(\theta)\neq 0. \] Define \[ Y_n(u)= Y_n(u,\theta)= \sum_{i=1}^n \ln(f(X_i,\theta+u/n)/ f(X_i,\theta)). \] It is known in the literature that, under certain conditions, the process \(Y_n(u)\) tends in distribution to a limiting process \[ Y(u)= u(p-q)x'+ (\nu_+(px'u)- \nu_-(-qx'u)) \ln(q/p) \] where \(\nu_\pm(u)\) are independent standard Poisson processes for \(u\geq 0\), extended by 0 to the negative half axis. Let \(u^*\) denote the random variable which maximizes the process \(Y(u)\). Let \(u_n^\pm\) denote the rightmost and leftmost maximizers of the empirical loglikelihood process \(Y_n(u)\). The paper investigates the convergence rates of the distributions of \(u_n^\pm\) to that of \(u^*\), under various conditions.