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

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:

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
\Bibitem{Bor18} \by A.~A.~Borovkov \paper On estimation of parameters in the case of discontinuous densities \jour Teor. Veroyatnost. i Primenen. \yr 2018 \vol 63 \issue 2 \pages 211--239 \mathnet{http://mi.mathnet.ru/tvp5141} \crossref{https://doi.org/10.4213/tvp5141} \elib{https://elibrary.ru/item.asp?id=32823078} \transl \jour Theory Probab. Appl. \yr 2018 \vol 63 \issue 2 \pages 169--192 \crossref{https://doi.org/10.1137/S0040585X97T98899X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000448195800001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85056951809} 

• http://mi.mathnet.ru/eng/tvp5141
• https://doi.org/10.4213/tvp5141
• http://mi.mathnet.ru/eng/tvp/v63/i2/p211

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. E. Mosyagin, “Exact asymptotics for the distribution of the time of attaining the maximum for a trajectory of a compound Poisson process with linear drift”, Siberian Adv. Math., 30:1 (2020), 26–42
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
3. V. E. Mosyagin, “Asimptotika raspredeleniya momenta dostizheniya maksimuma traektoriei protsessa Puassona so snosom i izlomom”, Teoriya veroyatn. i ee primen., 66:1 (2021), 94–109
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