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Projecting Statistical Functionals

  • Tomasz Rychlik

Part of the Lecture Notes in Statistics book series (LNS, volume 160)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Tomasz Rychlik
    Pages 1-9
  3. Tomasz Rychlik
    Pages 11-31
  4. Tomasz Rychlik
    Pages 33-54
  5. Tomasz Rychlik
    Pages 55-93
  6. Tomasz Rychlik
    Pages 95-129
  7. Tomasz Rychlik
    Pages 131-143
  8. Tomasz Rychlik
    Pages 145-155
  9. Tomasz Rychlik
    Pages 157-161
  10. Back Matter
    Pages 163-178

About this book

Introduction

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.

Keywords

Variance average probability statistics

Authors and affiliations

  • Tomasz Rychlik
    • 1
  1. 1.Institute of MathematicsPolish Academy of SciencesToruńPoland

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-2094-7
  • Copyright Information Springer-Verlag New York 2001
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-95239-0
  • Online ISBN 978-1-4612-2094-7
  • Series Print ISSN 0930-0325
  • Buy this book on publisher's site