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 P-theta of a sample X of size n has at least one point of discontinuity x(theta), x'(theta) not equal 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 (theta) over tilde of the parameter theta (possibly constructed from the same sample 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 P(theta(g)* - theta > v/n) and 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'minskil, 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 <(theta)over cap>i.e., P-theta(n(theta(g)* - theta) > v) similar to P-theta(n((theta) over cap - ) > v) for any v and n ->infinity; (3) the value of g can be chosen so that the inequality E-theta(theta(g)* - theta)(2) < E-theta(<(theta)over tilde> - theta)(2) is possible for sufficiently large n. Effectively no smoothness conditions are imposed on f(theta)(x). With an available "auxiliary" consistent estimator (theta) over tilde, simple rules are suggested for finding estimators theta(g)* which are asymptotically equivalent to (theta) over cap. The limiting distribution of n(theta(g)* - theta) as n -> infinity is studied.