Shrinkage Estimation

ISBN-13: 9783030021849 / Angielski / Twarda / 2018 / 333 str.

Dominique Fourdrinier; William E. Strawderman; Martin T. Wells

Opis

This book provides a coherent framework for understanding shrinkage estimation in statistics. The term refers to creating a new, more centralized estimate by shrinking an original raw estimate towards a truer mean. This results in more stable estimates for population parameters, reduced sampling and non-sampling errors, and smoothed spatial fluctuations. The book focuses primarily on point and loss estimation for the mean vector for multivariate normal and spherically symmetric distributions. Chapter 1 introduces the statistical and decision theoretic terminology and results that will be used throughout the book. Chapter 2 is concerned with estimating the p-dimensional mean vector of a multivariate normal distribution under quadratic loss from a frequentist perspective. In Chapter 3 the authors take a Bayesian view of shrinkage estimation. Chapter 4 introduces the general class of spherically symmetric distributions. Point estimation for this broad class is studied in subsequent chapters. In particular, Chapter 5 extends many of the results from Chapters 2 and 3 to spherically symmetric distributions. Chapter 6 considers the general linear model with spherically symmetric error distributions when a residual vector is available. Chapter 7 then considers the problem of estimating a location vector which is constrained to lie in a convex subset of Rp. Much of the chapter is devoted to one of two types of constraint sets, balls and polyhedral cones. In Chapter 8 the authors switch gears away from location parameter estimation and focus on loss estimation and data-dependent evidence reports. Appendices then cover Weakly Differentiable Functions; Examples of Weakly Differentiable Functions; Vanishing Of the Bracketed Term in Stein's Identity; Examples of Settings Where Stein's Identity Does Not Hold; Stein's Lemma and Stokes' Theorem for Smooth Boundaries; An Expression Of the Haff Operator; Harmonic, Superharmonic and Subharmonic Functions; Differentiation of Marginal Densities; Results on Expectation and Integrals; and Modified Bessel Functions.

Recenzje

"This book a timely and well-written exposition of shrinkage, or Stein, estimation intended for graduate students and researchers who wish to learn more about the topic." (Éric Marchand, Mathematical Reviews, August, 2019)
"The well-written volume, presenting the actual knowledge in this field, is suitable for readers having good background in analysis, linear algebra, probability theory and mathematical statistics." (Kurt Marti, zbMATH 1411.62011, 2019)

Spis treści

Chapter 1. Decision Theory Preliminaries.- Chapter 2. Estimation of a normal mean vector I.- Chapter 3. Estimation of a normal mean vector II.- Chapter 4. Spherically symmetric distributions.- Chapter 5. Estimation of a mean vector for spherically symmetric distributions I: known scale.- Chapter 6. Estimation of a mean vector for spherically symmetric distributions II: with a residual.- Chapter 7. Restricted Parameter Spaces.- Chapter 8. Loss and Confidence Level Estimation.-

O autorze

Dominique Fourdrinier is a Professor of Mathematical Statistics at the University of Rouen in France and an Adjunct Professor of Statistical Science at Cornell University. He earned his M.S. and Ph.D. degrees, both in Mathematical Statistics, at the University of Rouen. He is noted for his deep insights on the connections between shrinkage estimation and the properties of differential operators and has made important contributions to Bayesian statistics, decision theory, estimation theory, spherical and elliptical symmetry, the Stein phenomena as well as to statistical methods for signal and image processing.

 

William E. Strawderman is a Professor of Statistics at Rutgers University. He earned an M.S. in Mathematics from Cornell University and a second M.S. in Statistics from Rutgers, and then completed his Ph.D. in Statistics, also at Rutgers. He is a fellow of both the Institute of Mathematical Statistics and American Statistical Society and an Elected Member, International Statistical Institute. In 2015 he was named a Distinguished Alumni at Cornell. He is noted for path-breaking work in shrinkage estimation and has made fundamental contributions to a number of additional areas in statistics, including Bayesian statistics, decision theory, spherical symmetry, and biostatistics.

Martin T. Wells is the Charles A. Alexander Professor of Statistical Sciences at Cornell University. He is also a Professor of Social Statistics, Professor of Biostatistics and Epidemiology at Weill Cornell Medicine as well as an Elected Member of the Cornell Law School Faculty. He is a fellow of both the Institute of Mathematical Statistics and American Statistical Society and an Elected Member, International Statistical Institute. His research interests include Bayesian statistics, biostatistics, decision theory, empirical legal studies, machine learning, and statistical genomics.

Blurb

This book provides a coherent framework for understanding shrinkage estimation in statistics. The term refers to modifying a classical estimator by moving it closer to a target which could be known a priori or arise from a model. The goal is to construct estimators with improved statistical properties. The book focuses primarily on point and loss estimation of the mean vector of multivariate normal and spherically symmetric distributions. 

Chapter 1 reviews the statistical and decision theoretic terminology and results that will be used throughout the book. 

Chapter 2 is concerned with estimating the mean vector of a multivariate normal distribution under quadratic loss from a frequentist perspective. In Chapter 3 the authors take a Bayesian view of shrinkage estimation in the normal setting. Chapter 4 introduces the general classes of spherically and elliptically symmetric distributions. Point and loss estimation for these broad classes are studied in subsequent chapters. In particular, Chapter 5 extends many of the results from Chapters 2 and 3 to spherically and elliptically symmetric distributions. 

Chapter 6 considers the general linear model with spherically symmetric error distributions when a residual vector is available. Chapter 7 then considers the problem of estimating a location vector which is constrained to lie in a convex set. Much of the chapter is devoted to one of two types of constraint sets, balls and polyhedral cones. In Chapter 8 the authors focus on loss estimation and data-dependent evidence reports. 

Appendices cover a number of technical topics including weakly differentiable functions; examples where Stein’s identity doesn’t hold; Stein’s lemma and Stokes’ theorem for smooth boundaries; harmonic, superharmonic and subharmonic functions; and modified Bessel functions.