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 Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 1, Pages 126–144 (Mi tvp239)

Adaptive estimation of distribution density in the basis of algebraic polynomials

Institute of Mathematics and Informatics

Abstract: This paper is devoted to the problem of adaptive statistical estimation of the distribution density defined on a finite interval. Projective-type estimators in the basis of Jacobi polynomials is considered. An adaptive statistical estimator, which is asymptotically minimax in the case of mean-square losses for all sets from a certain family of contracting sets of functions having different smoothness, is constructed. The smoothness conditions are stated in terms of $L_2$-norms of residuals of distribution densities when approximating them by linear combinations of a finite number of the first Jacobi polynomials. Extension of the result to other orthonormal bases possessing some natural regularity properties is also discussed.

Keywords: adaptive estimation, locally minimax estimation, Jacobi polynomials, projective-type estimators, mean-square losses.

DOI: https://doi.org/10.4213/tvp239

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English version:
Theory of Probability and its Applications, 2005, 49:1, 93–109

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Revised: 28.05.2003
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Citation: R. Rudzkis, M. Radavicius, “Adaptive estimation of distribution density in the basis of algebraic polynomials”, Teor. Veroyatnost. i Primenen., 49:1 (2004), 126–144; Theory Probab. Appl., 49:1 (2005), 93–109

Citation in format AMSBIB
\Bibitem{RudRad04} \by R.~Rudzkis, M.~Radavicius \paper Adaptive estimation of distribution density in the basis of algebraic polynomials \jour Teor. Veroyatnost. i Primenen. \yr 2004 \vol 49 \issue 1 \pages 126--144 \mathnet{http://mi.mathnet.ru/tvp239} \crossref{https://doi.org/10.4213/tvp239} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2141333} \zmath{https://zbmath.org/?q=an:1089.62038} \transl \jour Theory Probab. Appl. \yr 2005 \vol 49 \issue 1 \pages 93--109 \crossref{https://doi.org/10.1137/S0040585X97980890} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000228185300006} 

• http://mi.mathnet.ru/eng/tvp239
• https://doi.org/10.4213/tvp239
• http://mi.mathnet.ru/eng/tvp/v49/i1/p126

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This publication is cited in the following articles:
1. Efromovich S., “Adaptive estimation of and oracle inequalities for probability densities and characteristic functions”, Annals of Statistics, 36:3 (2008), 1127–1155
2. Saadi N. Adjabi S. Gannoun A., “The Selection of the Number of Terms in An Orthogonal Series Cumulative Function Estimator”, Stat. Pap., 59:1 (2018), 127–152
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