Table Of ContentCONFIDENCE SETS FOR FUNCTIONS OF VARIANCE
COMPONENTS IN A MIXED LINEAR MODEL
BY
ROBERT M. BASKIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1991
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to Professor Malay
Ghosh for being my advisor, for proposing the problem of this
dissertation, and for guiding me to its conclusion. Words cannot
express what he has given to me. I would like to thank Professors
Ronald Randles, P. V. Rao, Andre Khuri and Joe Glover for serving on
my committee. Their efforts are truly appreciated. I would like to
thank Andrew Rosalsky, Myron Chang, Geoff Vining and Mark Yang for
their special attention and kindness.
I am indebted to Richard Scheaffer, Alan Agresti, Malay Ghosh,
Ronald Randles, and John Saw for guiding me through the graduate
program at the University of Florida. If it were not for Ronald
Randles I would not be here.
I am indebted to Mike Conlon, Art Smith, Phil Padgett and
everyone who made the departmental computer system operate. If it
were not for them the simulations would never have been completed.
My special thanks go to Carol Rozear, Nancy Pipkin, Leslie
Easom, Marilyn Saddler and Cindy Zimmerman for taking care of many of
the details and technical aspects of my stay here.
Last, but not least, I would like to thank my kind and loving
wife whose patience has been unbounded.
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TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iH
ABSTRACT vi
CHAPTERS
ONE INTRODUCTION 1
1.1 Literature Review 1
1.2 The Subject of This Dissertation 6
1.3 Matrix Notations 9
TWO GENERAL LINEAR MODEL WITH TWO VARIANCE H
COMPONENTS
2.1 Introduction 11
2.2 Henderson Estimates of Variance 16
2.3 Multivariate Central Limit Theorem 19
2.4 Jackknife Estimates 31
THREE GENERAL LINEAR MODEL WITH SEVERAL
VARIANCE COMPONENTS 81
3.1 Introduction 81
3.2 Multivariate Central Limit Theorem 83
3.3 Jackknife Estimates 95
FOUR HIERARCHICAL BAYES ESTIMATION OF THE
VARIANCE RATIO 123
4.1 Introduction 123
4.2 The Derivation of the Bayes Estimator 125
4.3 Jackknifed Estimator of the Asymptotic
Variance 135
FIVE RESULTS OF SIMULATIONS 144
5.1 Introduction 144
5.2 The Simulation Results for the
Unbalanced Model 144
5.3 The Simulation Results for the
Balanced Model 148
IV
SIX SUMMARY AND FUTURE RESEARCH . 155
6.1 Summary 155
6.2 Future Research 156
BIBLIOGRAPHY 157
BIOGRAPHICAL SKETCH 161
v
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
CONFIDENCE SETS FOR FUNCTIONS OF VARIANCE
COMPONENTS IN A MIXED LINEAR MODEL
By
Robert M. Baskin
August, 1991
Chairman: Dr. Malay Ghosh
Major Department: Statistics
In this dissertation a distribution-free approach to construction
of confidence sets is employed by using resampling techniques in a
general linear model including covariates. Initially only models with
two variance components are explored since this gives rise to the
common situation of estimating the ’’heritability ratio.” Some of the
widely used models in small area estimation including the nested error
regression model, random regression coefficients model, etc.
considered by earlier authors are seen to be special cases of this
model. The asymptotic properties of the jackknifed versions of the
Henderson III estimators as well as the jackknifed estimates of
variance are studied. This is extended to models with several random
components but still including covariates. In the situation of a
balanced one-way Anova model with covariates a hierarchical Bayes
estimator of the variance ratio is derived. The asymptotic
distribution of this estimator is derived under a frequentist model.
vi
The jackknife estimate of variance for this estimator is shown to be
consistent and used to construct confidence sets for the variance
ratio. Finally computer simulations of some of these models are used
to illustrate the features of the work.
vi1
CHAPTER ONE
INTRODUCTION
1.1 Literature Review
While the employment of linear models can be traced back for
centuries in specific fields such as astronomy, where they have been
used for predicting the position of celestial bodies, the latter half
of this century has seen linear models rise to be a standard method in
many other fields, especially such fields as genetic selection,
psychometrics, and survey sampling. Indeed, breeders of both plants
and animals have used variance components in linear models extensively
for predicting characteristics of future progeny since the landmark
works of Henderson (1950,1953). For purposes of selecting the most
desirable trait of progeny under consideration, breeders produce what
is referred to as a selection index based on certain linear
combinations of fixed and random components under a mixed linear
model. Discussion of selection indices in plant breeding may be found
in Henderson (1963) while a discussion of these indices in dairy
cattle breeding may be found in Lush and Shrode (1950) or Henderson,
Kempthorne, Searle, and von Krosigh (1959). Other references in this
area are cited in Gianola and Fernando (1986) and Harville (1990).
The fields of education and psychometrics have made varied use of
1
.
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linear models for measurement purposes. Further references are given
in Algina and Crocker (1986).
In survey sampling, survey analysts have used linear models in
finite population sampling to predict characteristics of the unsampled
units based on the observed sample. One of the first recognized uses
of linear models in general, and variance components in particular, in
the area of survey sampling is due to Cochran (1939). Because of
limited resources as well as rapid development of sophisticated
statistical techniques by such pioneers as Cochran, sample surveys
have been widely used in this country for over half a century. These
surveys have provided reliable statistics on the national and state
level with growing regularity and with the advent of electronic
telecommunications equipment have increased in both frequency and
application. However, in the past, the use of these methods in
sublevels below the state level has been limited because the estimates
for these sublevels have usually been based on small sample sizes, at
least within the sublevels, and thus these estimates produced
unacceptably large standard errors. Therefore, little of the early
work of survey analysts was devoted to producing reliable small area
estimators
In recent years, however, small area estimation has grown into an
important topic in survey sampling. Many government agencies such as
the United States Census Bureau, Statistics Canada, and the Central
Bureau of Statistics of Norway have been involved in estimating
population counts, unemployment rates, per capita income, etc. for
3
sublevels below the state, province, or fylker level. In this light,
small area estimation techniques have been devised that ’’borrow
strength” from similar neighboring areas for estimation and prediction
purposes. A good review of small area estimators may be found in
Ghosh and Rao (1991). This review shows how linear model techniques
can be used to address small area estimation problems.
In determining the effectiveness of the Henderson method of
estimating the variance components in a linear mixed model
statisticians have proposed certain minimal properties for a desirable
estimator. In an early paper by Yates and Zacopancy (1935) an example
is given showing the possibility of producing a negative estimate of
variance by a standard Anova estimate. With few exceptions (see for
example Smith and Murray (1984)) this particular facet of the problem
has always been a drawback from the point of view of most
statisticians. Typically the solution to this problem is to use the
positive part of the variance estimator as in Crump (1951) or Snedecor
and Cochran (1967). As shown by Pukelsheim (1977,1981) and discussed
further by Rao and Kleffe (1988) the situations in which linear
combinations of the effects in a linear model are nonnegatively
estimable are rare.
In a comparison of several variance estimators Corbeil and Searle
(1976) point out that maximum likelihood estimators of variance
components require that the likelihood function be maximized over the
positive space of the variance component parameters. In an extensive
work Harville (1977) reviews much of the material on standard Anova
