On estimation of parameters in the case of discontinuous densities

This paper is concerned with the problem of construction of estimators of parameters in the case when the density $f_\theta(x)$ of the distribution $\mathbf{P}_\theta$ of a sample $\mathrm X$ of size $n$ has at least one point of discontinuity $x(\theta)$, $x'(\theta)\neq 0$. It is assumed that either (a) from a priori considerations one can specify a localization of the parameter $\theta$ (or points of discontinuity) satisfying easily verifiable conditions, or (b) there exists a consistent estimator $\widetilde{\theta}$ of the parameter $\theta$ (possibly constructed from the same sample $\mathrm{X}$), which also provides some localization. Then a simple rule is used to construct, from the segment of the empirical distribution function defined by the localization, a family of estimators $\theta^*_{g}$ that depends on the parameter $g$ such that (1) for sufficiently large $n$, the probabilities $\mathbf{P}(\theta^*_{g}-\theta>v/n)$ and $\mathbf{P}(\theta^*_{g}-\theta<-v/n)$ can be explicitly estimated by a $v$-exponential bound; (2) in case (b) under suitable conditions (see conditions I–IV in Chap. 5 of [I. A. Ibragimov and R. Z. Has'minskiĭ, Statistical Estimation. Asymptotic Theory, Springer, New York, 1981], where maximum likelihood estimators were studied), a value of $g$ can be given such that the estimator $\theta^*_{g}$ is asymptotically equivalent to the maximum likelihood estimator $\widehat{\theta}$; i.e., $\mathbf{P}_\theta(n(\theta^*_{g}-\theta)>v)\sim \mathbf{P}_\theta(n(\widehat{\theta}-\theta)>v)$ for any $v$ and $n\to\infty$; (3) the value of $g$ can be chosen so that the inequality $\mathbf{E}_\theta(\theta^*_{g}-\theta)^2< \mathbf{E}_\theta(\widehat{\theta}-\theta)^2$ is possible for sufficiently large $n$. Effectively no smoothness conditions are imposed on $f_\theta(x)$. With an available “auxiliary” consistent estimator $\widetilde{\theta}$, simple rules are suggested for finding estimators $\theta^*_g$ which are asymptotically equivalent to $\widehat{\theta}$. The limiting distribution of $n(\theta^*_g-\theta)$ as $n\to\infty$ is studied.