Restricted Parameter Space Estimation Problems: Admissibility and Minimaxity Properties

ISBN-13: 9780387337470 / Angielski / Miękka / 2006 / 168 str.

Constance Van Eeden

Opis

This monograph is addressed to anyone interested in the subject of restrict- parameter-space estimation, and in particular to those who want to learn, or bring their knowledge up to date, about (in)admissibility and minimaxity problems for such parameter spaces. The coverage starts in the early 1950s when the subject of inference for - stricted parameter spaces began to be studied and ends around the middle of 2004. It presents known, and also some new, results on (in)admissibility and minimaxity for nonsequential point estimation problems in restricted ?ni- dimensional parameter spaces. Relationships between various results are d- cussed and open problems are pointed out. Few complete proofs are given, but outlines of proofs are often supplied. The reader is always referred to the published papers and often results are clari?ed by presenting examples of the kind of problems an author solves, or of problems that cannot be solved by a particular result. The monograph does not touch on the subject of testing hypotheses in - stricted parameter spaces. The latest books on that subject are by Robertson, Wright and Dykstra (1988) and Akkerboom (1990), but many new results in that area have been obtained since. The monograph does have a chapter in which questions about the existence of maximum likelihood estimators are discussed. Some of their properties are also given there as well as some algorithms for computing them. Most of these results cannot be found in the Robertson, Wright, Dykstra book.

Recenzje

From the reviews:

"This book is well written and well organized. The material is motivated with examples...[t]he monograph could be used for PhD-level special topics class...,[and] students with a background in mathematical statisics would benefit from researching and presenting in detail the specific examples of interesting problems." (Mary Meyer, for Biometrics, December 2007)

"This monograph contains a critical review of 50 years of research on estimators in convex, restricted k-dimensional parameter spaces … . The bibliography contains about 300 items, mostly published papers and books on the subject. … Questions about the existence and properties of maximum likelihood estimators are discussed, and some algorithms for computing them are presented. … The book represents a valuable guide to the literature in the area of restricted-space estimation and to what is known and not known in this area." (Jon Stene, Mathematical Reviews, Issue 2008 h)

"The literature on the estimation of parameters subject to order restrictions is not well documented in any book until now, and the author should be congratulated for writing an important book that fills the gap. The book provides an excellent summary of several major results available in the literature on the estimation of parameters subject to order restrictions. … the 16-page bibliography, covering over 250 publications, serves as an excellent reference for researchers interested in this field." (Shyamal D. Peddada, Journal of the American Statistical Association, Vol. 103 (482), June, 2008)

"The book under review is addressed to the audience interested in the subject of admissibility and minimaxity in restricted-parameter-space estimation. It covers results from the time when the subject began to be studied … to the middle of 2004, depicts relationships between various results, and points out open problems. … The introduction, which gives some history and examples, is followed by seven chapters. … An extensive bibliography is provided." (Joseph Melamed, Zentralblatt MATH, Vol. 1160, 2009)

Spis treści

Introduction: Some history and some examples.- A statement of the problem, the notation and some definitions.- (In)admissibility and dominators.- Minimax estimators and their admissibility.- Presence of nuisance parameters.- The linear model.- Other properties.- Existence of MLEs and algorithms to compute them.

Blurb

This monograph contains a critical review of 50 years of results on questions of admissibility and minimaxity of estimators of parameters that are restricted to closed convex subsets of R^k . It presents results of approximately 300 mostly-published papers on the subject, and points out relationships between them as well as open problems. The book does not touch on the subject of testing hypotheses for such parameter spaces. It does give an overview of known algorithms for computing maximum likelihood estimators under order-restrictions.

The book should be valuable as a reference for researchers and graduate students looking for what is known and unknown in the area of restricted parameter-space-estimation. It assumes a good knowledge of decision theory.

Constance van Eeden is Professeur émérite at the Université de Montréal, Honorary Professor at The University of British Columbia, and Professeure associée at the Université du Québec à Montréal. She previously held appointments at the Centrum voor Wiskunde en Informatica (1951–1960), Michigan State University (1960–1961), University of Minnesota (1961–1965), and Université de Montréal (1965–1989). She was a General Editor of Statistical Theory and Method Abstracts (1990–2004) and Associate Editor of the Annals of Statistics (1974–1977), The Canadian Journal of Statistics (1980–1994) and Annales des sciences mathématiques du Québec (1986–1998). She is a reviewer for Mathematical Reviews and a member of the Noether Award Committee. The Statistical Society of Canada awarded her their Gold Medal in 1990 and the Département de mathématiques et de statistique at the Université de Montréal named their yearly prize for the best-finishing undergraduate student in actuarial studies or statistics, the Prix Constance-van-Eeden. She is a Fellow of the Institute of Mathematical Statistics and of the American Statistical Association, and an Elected Member of the International Statistical Institute. She (co-)authored 66 papers in refereed journals, as well as two books and (co-)supervised 14 PhD and 19 MSc students.