Ebook: Projecting Statistical Functionals

About 10 years ago I began studying evaluations of distributions of or­ der statistics from samples with general dependence structure. Analyzing in [78] deterministic inequalities for arbitrary linear combinations of order statistics expressed in terms of sample moments, I observed that we obtain the optimal bounds once we replace the vectors of original coefficients of the linear combinations by the respective Euclidean norm projections onto the convex cone of vectors with nondecreasing coordinates. I further veri­ fied that various optimal evaluations of order and record statistics, derived earlier by use of diverse techniques, may be expressed by means of projec­ tions. In Gajek and Rychlik [32], we formulated for the first time an idea of applying projections onto convex cones for determining accurate moment bounds on the expectations of order statistics. Also for the first time, we presented such evaluations for non parametric families of distributions dif­ ferent from families of arbitrary, symmetric, and nonnegative distributions. We realized that this approach makes it possible to evaluate various func­ tionals of great importance in applied probability and statistics in different restricted families of distributions. The purpose of this monograph is to present the method of using pro­ jections of elements of functional Hilbert spaces onto convex cones for es­ tablishing optimal mean-variance bounds of statistical functionals, and its wide range of applications. This is intended for students, researchers, and practitioners in probability, statistics, and reliability.

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This monograph presents a general method of establishing explicit solutions to classical problems of calculating the best lower and upper mean-variance bounds on various statistical functionals over various nonparametric families of distributions. The functionals include quantiles, standard and conditional expectations of record and order statistics from independent and dependent samples, and a variety of their combinations important in statistics and reliability. The following families of distributions are taken into account: arbitrary, symmetric, symmetric unimodal, and U-shaped ones, distributions with monotone density and failure rate, and monotone density and failure rate on the average distributions. The method is based on determining projections of the functionals onto properly chosen convex cones in functional Hilbert spaces. It allows us to explicitly point out the distributions which attain the bounds. The book is addressed to students, researchers, and practitioners in statistics and applied probability. Most of the results have been established recently, and a significant part of them has not been published yet. Numerous open problems are stated in the text.

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This monograph presents a general method of establishing explicit solutions to classical problems of calculating the best lower and upper mean-variance bounds on various statistical functionals over various nonparametric families of distributions. The functionals include quantiles, standard and conditional expectations of record and order statistics from independent and dependent samples, and a variety of their combinations important in statistics and reliability. The following families of distributions are taken into account: arbitrary, symmetric, symmetric unimodal, and U-shaped ones, distributions with monotone density and failure rate, and monotone density and failure rate on the average distributions. The method is based on determining projections of the functionals onto properly chosen convex cones in functional Hilbert spaces. It allows us to explicitly point out the distributions which attain the bounds. The book is addressed to students, researchers, and practitioners in statistics and applied probability. Most of the results have been established recently, and a significant part of them has not been published yet. Numerous open problems are stated in the text.
Content:
Front Matter....Pages i-ix
Introduction and Notation....Pages 1-9
Basic Notions....Pages 11-31
Quantiles....Pages 33-54
Order Statistics of Independent Samples....Pages 55-93
Order Statistics of Dependent Observations....Pages 95-129
Records and kth Records....Pages 131-143
Predictions of Order and Record Statistics....Pages 145-155
Further Research Directions....Pages 157-161
Back Matter....Pages 163-178

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This monograph presents a general method of establishing explicit solutions to classical problems of calculating the best lower and upper mean-variance bounds on various statistical functionals over various nonparametric families of distributions. The functionals include quantiles, standard and conditional expectations of record and order statistics from independent and dependent samples, and a variety of their combinations important in statistics and reliability. The following families of distributions are taken into account: arbitrary, symmetric, symmetric unimodal, and U-shaped ones, distributions with monotone density and failure rate, and monotone density and failure rate on the average distributions. The method is based on determining projections of the functionals onto properly chosen convex cones in functional Hilbert spaces. It allows us to explicitly point out the distributions which attain the bounds. The book is addressed to students, researchers, and practitioners in statistics and applied probability. Most of the results have been established recently, and a significant part of them has not been published yet. Numerous open problems are stated in the text.
Content:
Front Matter....Pages i-ix
Introduction and Notation....Pages 1-9
Basic Notions....Pages 11-31
Quantiles....Pages 33-54
Order Statistics of Independent Samples....Pages 55-93
Order Statistics of Dependent Observations....Pages 95-129
Records and kth Records....Pages 131-143
Predictions of Order and Record Statistics....Pages 145-155
Further Research Directions....Pages 157-161
Back Matter....Pages 163-178
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