Projecting Statistical Functionals
Published on 20. April 2001
Book
Paperback/Softback
IX, 175 pages
978-0-387-95239-0 (ISBN)
Description
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.
More details
Series
Edition
Softcover reprint of the original 1st ed. 2001
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
IX, 175 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 11 mm
Weight
300 gr
ISBN-13
978-0-387-95239-0 (9780387952390)
DOI
10.1007/978-1-4612-2094-7
Schweitzer Classification
Person
Professor N. Balakrishnan is a Distinguished University Professor in the Department of Mathematics and Statistics at McMaster University, Hamilton, Canada. He is a Fellow of the Royal Society of Canada, a Fellow of the American Statistical Association, a Fellow of the Institute of Mathematical Statistics, and a recipient of an honorary doctoral degree from the National and Kapodistrian University of Athens, Greece. He is an author of many well-known books and research monographs in Statistics including the well-known four encyclopedic volumes on Statistical Distributions published by John Wiley & Sons. He is currently the Editor-in-Chief of three series of Communications in Statistics as well as Mathematical Methods of Statistics.
Professor Tomasz Rychlik received the M.Sc. Degree in applied mathematics at the Faculty of Applied Mathematics and Technical Physics of the Technical University of Warsaw, Poland, in 1982. He defended his PhD thesis on robust statistics at the Institute of Mathematics, Polish Academy of Sciences in 1986. Now he is Professor of Mathematics at the Institute of Mathematics, Polish Academy of Sciences. His research interests contain probabilistic inequalities, reliability theory, and stochastic orders. He has published over 120 papers. Most of them were devoted to inequalities for ordered random variables.
Content
1 Introduction and Notation.- 1.1 Introduction.- 1.2 Notation.- 2 Basic Notions.- 2.1 Elements of Hilbert Space Theory.- 2.2 Statistical Linear Functionals.- 2.3 Restricted Families of Distributions.- 3 Quantiles.- 3.1 General and Symmetric Distributions.- 3.2 Distributions with Monotone Density and Failure Rate.- 3.3 Distributions with Monotone Density and Failure Rate on the Average.- 3.4 Symmetric Unimodal Distributions.- 3.5 Open Problems.- 4 Order Statistics of Independent Samples.- 4.1 General and Symmetric Distributions.- 4.2 Life Distributions with Decreasing Density and Failure Rate.- 4.3 Distributions with Monotone Density and Failure Rate on the Average.- 4.4 Symmetric Unimodal Distributions.- 4.5 Bias of Quantile Estimates.- 4.6 Open Problems.- 5 Order Statistics of Dependent Observations.- 5.1 Dependent Observations with Given Marginal Distribution.- 5.2 General and Symmetric Distributions.- 5.3 Distributions with Monotone Density and Failure Rate.- 5.4 Distributions with Monotone Density and Failure Rate on the Average.- 5.5 Symmetric Unimodal and U-Shaped Distributions.- 5.6 Bias of Quantile Estimates.- 5.7 Extreme Effect of Dependence.- 5.8 Open Problems.- 6 Records and kth Records.- 6.1 Dependent Identically Distributed Observations.- 6.2 General and Symmetric Distributions.- 6.3 Life Distributions with Decreasing Density and Failure Rate.- 6.4 Increments of Records.- 6.5 Open Problems.- 7 Predictions of Order and Record Statistics.- 7.1 General Distributions.- 7.2 Distributions with Decreasing Density and Failure Rate.- 7.3 Open Problems.- 8 Further Research Directions.- References.- Author Index.