Table Of ContentSPRINGER BRIEFS IN STATISTICS
Gauss M. Cordeiro
Francisco Cribari-Neto
An Introduction
to Bartlett
Correction and
Bias Reduction
SpringerBriefs in Statistics
For furthervolumes:
http://www.springer.com/series/8921
Gauss M. Cordeiro Francisco Cribari-Neto
•
An Introduction
to Bartlett Correction
and Bias Reduction
123
GaussM.Cordeiro
Francisco Cribari-Neto
Departamento de Estatística
Universidade Federalde Pernambuco
Recife, Pernambuco
Brazil
ISSN 2191-544X ISSN 2191-5458 (electronic)
ISBN 978-3-642-55254-0 ISBN 978-3-642-55255-7 (eBook)
DOI 10.1007/978-3-642-55255-7
Springer Heidelberg NewYork Dordrecht London
LibraryofCongressControlNumber:2014938215
MathematicsSubjectClassification(2010):62F03,62F10
(cid:2)TheAuthor(s)2014
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Preface
Statistical inference is oftentimes based on first-order asymptotic theory.
In particular, it is a common practice to perform likelihood ratio, score and Wald
tests using approximate critical values. Such critical values are obtained from the
test statistic limiting distribution when the null hypothesis is true. The approx-
imation holds when the number of observations in the sample tends to infinity,
and it is thus expected to deliver reliable inferences in large samples. When the
sample is not large, however, size distortions are likely to arise. That is, the
effective type Ierror probability may not be close to the nominal size selected by
the practitioner. It is thus important tohave at hand alternatives thatdeliver more
reliable inference in small samples. In this monograph, we cover analytical
corrections known as Bartlett and Bartlett-type corrections. Bartlett corrections
are applied to likelihood ratio test statistics whereas Bartlett-type corrections are
applied to score test statistics and also to other asymptotically v2 criteria.
The corrections deliver modified tests with error rates that decay faster toward
zero. Thus, such tests can be expected to display superior finite sample behavior.
Practitioners are usually also interested in parameter estimation. Maximum
likelihood estimators are typically consistent and asymptotically normal, but are
usually biased. That is, the estimator expected value differs from the true
parameter value, which implies the existence of a systematic error. We provide
analytical and numerical approaches that can be used to reduce the bias of the
maximum likelihood estimator. Preventive and corrective bias reduction schemes
are presented and discussed. The former entails modifying the log likelihood or
score function priortoobtainingthe estimator whereasthe latter entails obtaining
theestimatorintheusualfashionandthenbiascorrectingit.Thesebiascorrections
candeliver modifiedestimatorsthathavemuchsmallersystematicerrors thanthe
corresponding unmodified estimators.
The material we present in this book is a compilation of analytical results
andnumericalevidenceavailableintheliterature.Wedonotincludenewresults.
Our goal is to present, in a coherent way, strategies that can be used to achieve
more accurate inferences. Our main focus lies on obtaining analytical corrections
to tests that are based on a first-order asymptotic approximation and also on
achieving bias reduction in small samples. Numerical (Monte Carlo) results are
presented in order to illustrate the gains involved in using such finite sample
vii
viii Preface
corrections.Wealsopointoutthatthedetailsinvolvedinmanyofthederivations
were not included in the text since we intend to provide readers with a concise
monograph.Furtherdetailscanbefoundinthereferenceslistedattheendofeach
chapter.
The structure of our monograph was dictated by three main choices we made.
First, we intend to provide readers with a concise overview of the topic. Second,
we decided to focus on point estimation and testing inference. We do so by
focusingonbiasreductionofestimatorsandcorrectionsthatcanbeappliedtotest
statistics. Additionally, even though our focus lies on analytical corrections we
also include material on bootstrap-based inference since it is often cited as an
appealing alternative to analytically corrected estimators and tests.
Finally, we would like to thank Klaus Vasconcellos for suggestions on parts
of our monograph. We also gratefully acknowledge financial support from
CNPq/Brazil.
Recife, February 2014 Gauss M. Cordeiro
Francisco Cribari-Neto
Contents
1 Likelihood-Based Inference and Finite-Sample Corrections:
A Brief Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Likelihood Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Some Properties of Maximum Likelihood Estimators. . . . . . . . 3
1.4 A Simple Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Likelihood-Based Testing Inference. . . . . . . . . . . . . . . . . . . . 5
1.6 Some Remarks on Bartlett and Bartlett-Type Corrections. . . . . 6
1.7 Some Remarks on Bias Corrections. . . . . . . . . . . . . . . . . . . . 8
1.8 Some Remarks on the Bootstrap . . . . . . . . . . . . . . . . . . . . . . 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Bartlett Corrections and Bootstrap Testing Inference . . . . . . . . . . 13
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Bartlett Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Lawley’s Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Bartlett-Corrected Likelihood Ratio Tests. . . . . . . . . . . . . . . . 20
2.5 Generalized Linear Models. . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Bartlett Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.2 Special Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.3 Computer Codes for Calculating Bartlett Corrections . . . 29
2.6 Birnbaum–Saunders Non-linear Regression Models . . . . . . . . . 32
2.7 Bootstrap-Based Hypothesis Testing. . . . . . . . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Bartlett-Type Corrections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Bartlett-Type Correction to the Score Statistic. . . . . . . . . . . . . 46
3.3 An Extended Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Barttlett-Type Correction to the Wald Statistic . . . . . . . . . . . . 51
3.5 One-Parameter Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6 The p(cid:2) Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
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3.7 Generalized Linear Models. . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.8 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.9 Heteroskedastic Regression. . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Analytical and Bootstrap Bias Corrections . . . . . . . . . . . . . . . . . . 69
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 A General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 One-Parameter Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Two-Parameter Distributions. . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Generalized Linear Models. . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 The Birnbaum–Saunders Model. . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Special Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.8 Monte Carlo Simulation Evidence . . . . . . . . . . . . . . . . . . . . . 83
4.9 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.10 Linear Heteroskedastic Regression. . . . . . . . . . . . . . . . . . . . . 86
4.11 Beta Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.12 An Alternative Analytical Bias Correction . . . . . . . . . . . . . . . 93
4.13 Bootstrap Bias Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Appendix A: Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . 101
Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Acronyms
BS Birnbaum–Saunders
GLM Generalized linear model
LBC Linear-bias-correction
LR Likelihood ratio
ML Maximum likelihood
MLE Maximum likelihood estimator
OLS Ordinary least squares
OLS Ordinary least squares estimator
SAR Synthetic aperture radar
xi
