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Teoriya Veroyatnostei i ee Primeneniya, 2018, Volume 63, Issue 3, Pages 545–564
DOI: https://doi.org/10.4213/tvp5174 (Mi tvp5174) 		 
Exact bounds on the truncated-tilted mean, with applications

I. Pinelis

Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, USA
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DOI: https://doi.org/10.4213/tvp5174
Abstract: Exact upper bounds for ${\mathbf{E} Xe^{h(X\wedge w)}}/{\mathbf{E} e^{h(X\wedge w)}}$, which is the expectation of the Cramér transform of the so-called Winsorized-tilted mean of a random variable, are given in terms of its first two moments. Such results are needed in work with nonuniform Berry–Esseen-type bounds for general nonlinear statistics. As another application, optimal upper bounds on the Bayes posterior mean are provided. Certain monotonicity properties of the tilted mean are also presented.
Keywords: exact bound, Winsorization, truncation, large deviation, nonuniform Berry–Esseen-type bounds, Cramér transform, monotonicity, Bayes posterior mean.
Received: 05.04.2016
Revised: 17.04.2018
Accepted: 20.12.2017
English version:
Theory of Probability and its Applications, 2019, Volume 63, Issue 3, Pages 447–463
DOI: https://doi.org/10.1137/S0040585X97T989167
Bibliographic databases:
Document Type: Article
Language: English
Citation: I. Pinelis, “Exact bounds on the truncated-tilted mean, with applications”, Teor. Veroyatnost. i Primenen., 63:3 (2018), 545–564; Theory Probab. Appl., 63:3 (2019), 447–463
Citation in format AMSBIB
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