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Empirical Likelihood




ISBN 9781584880714
Published May 18, 2001 by Chapman and Hall/CRC
304 Pages 40 B/W Illustrations

 
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Book Description

Empirical likelihood provides inferences whose validity does not depend on specifying a parametric model for the data. Because it uses a likelihood, the method has certain inherent advantages over resampling methods: it uses the data to determine the shape of the confidence regions, and it makes it easy to combined data from multiple sources. It also facilitates incorporating side information, and it simplifies accounting for censored, truncated, or biased sampling.

One of the first books published on the subject, Empirical Likelihood offers an in-depth treatment of this method for constructing confidence regions and testing hypotheses. The author applies empirical likelihood to a range of problems, from those as simple as setting a confidence region for a univariate mean under IID sampling, to problems defined through smooth functions of means, regression models, generalized linear models, estimating equations, or kernel smooths, and to sampling with non-identically distributed data. Abundant figures offer visual reinforcement of the concepts and techniques. Examples from a variety of disciplines and detailed descriptions of algorithms-also posted on a companion Web site at-illustrate the methods in practice. Exercises help readers to understand and apply the methods.

The method of empirical likelihood is now attracting serious attention from researchers in econometrics and biostatistics, as well as from statisticians. This book is your opportunity to explore its foundations, its advantages, and its application to a myriad of practical problems.

Table of Contents

EMPIRICAL LIKELIHOOD (EL)
Introduction
Empirical Distribution Function
Nonparametric Maximum Likelihood
Nonparametric Likelihood Ratios
Ties in the Data
Multinomial on the Sample
EL for a Univariate Mean
Coverage Accuracy
Power and Efficiency
Empirical versus Parametric Inferences
Computing the Empirical Likelihood
EL FOR RANDOM VECTORS
NPMLE for Random Vectors
EL for a Multivariate Mean
Fisher, Bartlett, and Bootstrap Calibration
Smooth Functions of Means
Estimating Equations
Transformation Invariance of EL
Using Side Information
Convex dual Problem
Unconstrained Dual Problem
Solving the Dual Problem
Euclidean Likelihood
Other Nonparametric Likelihoods
REGRESSION AND MODELING
Sampling Pairs
Fixed Regressors
Triangular Array ELT
Analysis of Variance
Variance Modeling
Nonlinear Least Squared
Generalized Linear Models
Generalized Projection Pursuit
Plastic pipe Data
Euclidean likelihood for Regression and ANOVA
SYMMETRY AND INDEPENDENCE
Testing Symmetry
Constraining to Symmetry
Approximate Symmetry
Symmetric Unimodal Distributions
Testing Independence
Constraining to Independence
Approximate Independence
Permutation Tests
IMPERFECTLY OBSERVED DATA
Biased Sampling
Truncation
Multiple Biased Samples
Censoring
CURVE ESTIMATION
Kernel Estimates
Bias and Variance
EL for Kernel Smooths
Blood Pressure Trajectories
Simultaneous Inference
Bands for the ECDF
Bands for the Quantile Function
DEPENDENT DATA
Reducing to Independence
Blockwise Empirical Likelihood
Hierarchical Data
Dual likelihood for Martingales
HYBRIDS AND CONNECTIONS
Parametric Models for Subsets of Data
Parametric Models for Components of the Data
Parametric Models for Data Ranges
Empirical Likelihood and Bayes
Bayesian Bootstrap
Nonparametric tilting Bootstrap
Weighted Likelihood Bootstrap
Bootstrap Likelihoods
Jackknifes
SOME PROOFS
Lemmas
Vector ELT
Triangular Array ELT
Multisample ELT
ALGORITHMS
Smooth Optimization
Simple Hypotheses
Composite Hypotheses
Overdetermined NPMLE
Constraints
Partial Derivatives
Nested Algorithms
Gradient Equations
Primal Problem
Sequential Linearization
Sequential Linearization and Estimating Equations
Semi-infinite Programming
Profiling Empirical Likelihoods
HIGHER ORDER ASYMPTOTICS
Bartlett Correction
Pseudo-Likelihood Theory
Signed Root Corrections
Least Favorable Families
Large Deviations

...
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Reviews

"In this beautifully written book Owen lucidly illustrates the wide applicability of empirical likelihood and provides masterful accounts of its latest theoretical developments. Numerous empirical examples should fascinate practitioners in various fields of science. I recommend this book extremely highly."
-Yuichi Kitamura, Department of Economics, University of Pennsylvania

"The statistical model discovery and information recovery process is shrouded in a great deal of uncertainty. Owen's empirical likelihood procedure provides an attractive basis for how best to represent the sampling process and to carry through the estimation and inference objectives"
- George Judge, University of California, Berkeley

"A great amount of thought and care has gone into preparing this fascinating monograph. Empirical likelihood is somehow at the junction between two of the main streams of contemporary statistics, parametric and nonparametric methods. Through EL, some of the key results of the former (such as Wilks' Theorem and Bartlett correctibility) carry over to the latter in a way which seems almost to deny the infinite-parameter character of nonparametric statistics. Even if the purpose of empirical likelihood was no more than this didactic one, it would be significant. Yet as Owen shows so engagingly, EL also has a colourful life of its own. It is a unique practical tool, and it enjoys important, and growing, connections to many areas of statistics, from the Kaplan-Meier estimator to the bootstrap and beyond. If we look at statistics from the vantage point of EL we can see a long way; Owen shows us how, and how far."
-Professor Peter Hall, Australian National University.

"This impressive monograph is the definitive source for researchers who wish to learn how to utilize empirical likelihood methods. The author addresses a range of topics, including univariate confidence intervals, regression models, kernel smoothing, and mean function smoothing. Although the book covers considerable ground and is rigorous, the book is well written and a reader with a solid background in mathematical statistics can readily tackle this volume."
-Journal of Mathematical Psychology

This book will make accessible to a wider audience the new and important area of nonparameteric likelihood and hypothesis testing. Masterfully written by a pioneer in this area, this book lucidly discusses the statistical theory and -- perhaps more importantly for applied econometricians -- computational details and practical aspects of putting the ideas to work with real data. This book will have a major impact on the way hypothesis testing is done in econometrics, where one is very often unsure about what the correct model specification is.
-Anand V. Bodapati, UCLA Anderson School of Management, USA

"The book will make an ideal text for a course in empirical likelihood for advanced statistics students, while it provides theoretically-minded practitioners a quick access to the growing empirical likelihood literature... The writing style is extremely clear throughout, even when discussing the fine points of the theory. Important results are well motivated, discussed and illustrated by real data examples."
-Biometrics, vol. 57, no. 4, December 2001