On estimation of parameters in the case of discontinuous densities

A. A. Borovkov

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1 Citation (Scopus)


This paper is concerned with the problem of construction of estimators of parameters in the case when the density fθ (x) of the distribution Pθ of a sample X of size n has at least one point of discontinuity x(θ), x(θ) ≠ 0. It is assumed that either (a) from a priori considerations one can specify a localization of the parameter θ (or points of discontinuity) satisfying easily verifiable conditions, or (b) there exists a consistent estimator (formula presented) of the parameter θ (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 θg that depends on the parameter g such that (1) for sufficiently large n, the probabilities P(θg − θ > v/n)and P(θ g − θ < −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 θg is asymptotically equivalent to the maximum likelihood estimator (formula presented) for any v and n → ∞; (3) the value of g can be chosen so that the inequality (formula presented) is possible for sufficiently large n. Effectively no smoothness conditions are imposed on fθ (x). With an available “auxiliary” consistent estimator (formula presented), simple rules are suggested for finding estimators θg which are asymptotically equivalent to(formula presented). The limiting distribution of n(θg −θ) as n → ∞ is studied.

Original languageEnglish
Pages (from-to)169-192
Number of pages24
JournalTheory of Probability and its Applications
Issue number2
Publication statusPublished - 1 Jan 2018


  • Change-point problem
  • Distribution with discontinuous density
  • Estimators of parameters
  • Infinitely divisible factorization
  • Maximum likelihood estimator
  • change-point problem
  • maximum likelihood estimator
  • infinitely divisible factorization
  • distribution with discontinuous density
  • estimators of parameters

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