Restricted Parameter Space Estimation Problems: Admissibility and Minimaxity Properties (Lecture Notes in Statistics)

172 Pages·2006·1.14 MB·English

by  Constance van Eeden

Lecture Notes in Statistics 188
Edited by P.Bickel,P.Diggle,S.Fienberg,U.Gather,
I.Olkin,S.Zeger
Constance van Eeden
Restricted Parameter Space
Estimation Problems
Admissibility and Minimaxity Properties
Constance van Eeden
Moerland 19
1151 BH Broek in Waterland
The Netherlands
[email protected]
Library ofCongress Control Number:2006924372
ISBN-10:0-387-33747-4
ISBN-13:978-0387-33747-0
Printed on acid-free paper.
© 2006 Springer Science+Business Media,LLC
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Restricted-Parameter-Space
Estimation Problems
Admissibility and Minimaxity Properties
Constance van Eeden
To the memory of Jan Hemelrijk (1918–2005),
who, in the early 1950s, encouraged me to work
on restricted-parameter-spaces-inference problems
and was very supportive during the preparation
of my 1958 PhD thesis on the subject.
Preface
Thismonographisaddressedtoanyoneinterestedinthesubjectofrestricted-
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 re-
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 finite-
dimensional parameter spaces. Relationships between various results are dis-
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 clarified 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 re-
strictedparameterspaces.ThelatestbooksonthatsubjectarebyRobertson,
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
alsogiventhereaswellassomealgorithmsforcomputingthem.Mostofthese
results cannot be found in the Robertson, Wright, Dykstra book.
The author’s long familiarity with the subject of this monograph combined
with her 14-year General Editorship of Statistical Theory and Method Ab-
stracts make it very unlikely that she missed any published results on the
subject.
viii
Theauthorthanksherco-authorandgoodfriendJimZidekforproposingshe
write a monograph on estimation in restricted parameter spaces and for his
encouragement and help during the writing.
A large part of the work on this monograph was done while the author was
spending fall semesters at the Department of Statistics at The University of
British Columbia. Many thanks for the support and the hospitality.
Also, many thanks to the librarians of CWI (Centrum for Wiskunde en In-
formatica, Amsterdam, The Netherlands) for their help in locating copies of
needed publications.
Finally,mythankstoseveralanonymousreviewersfortheirhelpfulcomments,
as well as to the staff at Springer’s editorial and production departments – in
particular to John Kimmel, Statistics Editor, – for their help in getting this
monograph in final form.
Broek in Waterland, June 2006
Contents
1 Introduction: Some history and some examples ............ 1
2 A statement of the problem, the notation and some
definitions ................................................. 5
3 (In)admissibility and dominators .......................... 15
3.1 Results for the exponential family ......................... 15
3.2 Results of Moors ........................................ 18
3.3 Results of Charras and van Eeden ......................... 20
3.4 Dominators............................................. 23
3.5 Universal domination and universal admissibility ............ 29
3.6 Discussion and open problems............................. 29
4 Minimax estimators and their admissibility ................ 33
4.1 Some helpful results ..................................... 33
4.2 Minimax results when Θ is bounded ....................... 40
4.3 Minimax results when k =1 and Θ is not bounded .......... 49
4.4 Minimax results when k >1 and Θ is not bounded .......... 59
4.5 Discussion and open problems............................. 63
5 Presence of nuisance parameters........................... 69
5.1 Location parameter estimation with known ν ............... 71
5.2 Scale parameter estimation with known ν................... 80
5.3 Unknown ν ............................................. 88
5.4 Polygonal, orthant and circular cones ...................... 96
5.5 (Admissible) minimax estimators .......................... 98
5.6 Discussion and open problems.............................100
6 The linear model ..........................................103
6.1 One linear inequality restriction ...........................104
6.2 Interval and quadrant restrictions .........................109
x Contents
6.3 Results of Moors and van Houwelingen for polyhedral and
ellipsoidal restrictions ...................................112
6.4 Discussion and open problems.............................112
7 Other properties...........................................115
7.1 Ideas from weighted likelihood estimation...................115
7.2 Robustness, gains in risk function and bias .................117
8 Existence of MLEs and algorithms to compute them.......131
8.1 The model and the conditions.............................131
8.2 Algorithms .............................................133
8.3 Norm-reducing properties of MLEs ........................139
8.4 Algorithms for multivariate problems ......................141
Bibliography...................................................143
Author Index..................................................161
Subject Index .................................................165