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Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 2, Pages 211–239 (Mi tvp5141)  

This article is cited in 2 scientific papers (total in 2 papers)

On estimation of parameters in the case of discontinuous densities

A. A. Borovkovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University

Abstract: 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.

Keywords: estimators of parameters, maximum likelihood estimator, distribution with discontinuous density, change-point problem, infinitely divisible factorization.

DOI: https://doi.org/10.4213/tvp5141

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English version:
Theory of Probability and its Applications, 2018, 63:2, 169–192

Bibliographic databases:

Received: 23.03.2017
Revised: 03.04.2017
Accepted:29.08.2017

Citation: A. A. Borovkov, “On estimation of parameters in the case of discontinuous densities”, Teor. Veroyatnost. i Primenen., 63:2 (2018), 211–239; Theory Probab. Appl., 63:2 (2018), 169–192

Citation in format AMSBIB
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\by A.~A.~Borovkov
\paper On estimation of parameters in the case of discontinuous densities
\jour Teor. Veroyatnost. i Primenen.
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\issue 2
\pages 211--239
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\crossref{https://doi.org/10.4213/tvp5141}
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\transl
\jour Theory Probab. Appl.
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\vol 63
\issue 2
\pages 169--192
\crossref{https://doi.org/10.1137/S0040585X97T98899X}
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    2. I. G. Kazantsev, B. O. Mukhametzhanova, K. T. Iskakov, T. Mirgalikyzy, “Vydelenie uglovykh struktur na izobrazheniyakh s pomoschyu masshtabiruemykh masok”, Sib. zhurn. industr. matem., 23:1 (2020), 70–83  mathnet  crossref
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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