Restricted Parameter Space Estimation Problems

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.

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, isfollowed by seven chapters. … An extensive bibliography is provided." (Joseph Melamed, Zentralblatt MATH, Vol. 1160, 2009)