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Cencov's comments; inverse problems in distribution estimation; $L_1$ density estimation; variational distance; $\phi$-divergence
N. N. Cencov wrote a commentary chapter included in the Appendix of the Russian translation of the Devroye and Györfi book [15] collecting some arguments supporting the $L_1$ view of density estimation. The Cencov's work is available in Russian only and it hasn't been translated, so late Igor Vajda decided to translate the Cencov's paper and to add some remarks on the occasion of organizing the session “25 Years of the $L_1$ Density Estimation” at the Prague Stochastics 2010 Symposium. In this paper we complete his task, i. e., we translate the Cencov's chapter and insert some remarks on the related literature focusing primarily on Igor's results. We would also like to acknowledge the excellent work of Alexandre Tsybakov who translated the Devroye and Györfi book in Russian, annotated it with valuable comments and included some related references published in Russian only.
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