Linear Models

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Overview

Linear models, normally presented in a highly theoretical and mathematical style, are brought down to earth in this comprehensive textbook. Linear Models examines the subject from a mean model perspective, defining simple and easy-to-learn rules for building mean models, regression models, mean vectors, covariance matrices and sums of squares matrices for balanced and unbalanced data sets. The author includes both applied and theoretical discussions of the multivariate normal distribution, quadratic forms, maximum likelihood estimation, less than full rank models, and general mixed models. The mean model is used to bring all of these topics together in a coherent presentation of linear model theory.

Key Features
• Provides a versatile format for investigating linear model theory, using the mean model
• Uses examples that are familiar to the student:
• design of experiments, analysis of variance, regression, and normal distribution theory
• Includes a review of relevant linear algebra concepts
• Contains fully worked examples which follow the theorem/proof presentation

Audience: Graduate students in statistics.

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Editorial Reviews

From the Publisher
"At the theorectical level, this book deals with the general linear model: the usual results on the distribution of linear functions of the observations and of quadratic forms are all derived in the general case."
—MATHEMATICAL REVIEWS
"This text presents the linear model (i.e., the analysis of variance and regression theory) from a sophisticated matrix algebra formulation. The book would be most suitable for graduate students of statistics who are already familiar with both linear algebra and the linear model."
—JOURNAL OF MATHEMATICL PSYCHOLOGY
Booknews
A textbook for graduate students with some previous exposure to mathematical statistics, linear algebra, normal distribution theory, linear regression, and the design of experiments. Rather than survey all aspects of linear models, focuses on a series of introductory topics to provide a solid foundation for further study. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

Meet the Author

Professor William Moser is a Professor of Chemical Engineering at the Worcester Polytechnic Institute. He is also a member of the Center for Inorganic Membrane Studies at WPI. Professor Moser invented both the Cylindrical Internal Reflectance Reactors (CIR-REACTORS) and Optical Fiber coupled CIR-Reactors (OFCIR-REACTORS), which are now commercial products used for reaction monitoring in homogeneous and heterogeneous catalysis. He was recently issued a patent on remote infrared sensing using optical fiber cables connected to high pressure reactors as well as normal laboratory glassware reactors. Dr. Moser is a Fellow of the New York Academy of Sciences, and is a co-founder of the Organic Reactions Catalysis Society. He is a past Chairman of the ACS Petroleum Division. He is the editor of several books on homogeneous and heterogeneous catalysis, and has a variety of publications and patents in the catalytic and materials science fields.

Zbynek Šidák was Chairman, Department of Probability and Statistics at the Mathematical Institute, Academy of Sciences, Czech Republic. He is now the principal research worker there. He has worked at various American universities as well. For 30 years, he was Editor of the journal Applications of Mathematics. His interests in statistics were rank tests, multivariate and cluster analysis, ranking and selection procedures, and Markov chains.

Pranab K. Sen is Cary C. Boshamer Professor of Biostatistics and Statistics at the University of North Carolina, and is a Fellow of the Institute of Mathematical Statistics and of the American Statistical Association. He is also an elected member of the International Statistical Institute.Prenab K. Sen is author or co-author of multiple volumes in Mathematical Statistics, Probability Theory and Biostatistics, and has published extensively in nonparametrics, multivariate and sequential analysis, and reliability and survival analysis.

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Table of Contents

Linear Algebra and Related Introductory Topics: Elementary Matrix Concepts. Kronecker Products. Random Vectors. Multivariate Normal Distribution: Multivariate Normal Distribution Function. Conditional Distributionsof Multivariate Normal Vectors. Distributions of Certain Quadratic Forms. Distributions of Quadratic Forms: Quadratic Forms of Normal Random Vectors. Independence. t and F Distributions. Bhats Lemma. Complete, Balanced Factorial Experiments: Models That Admit Restrictions (Finite Models). Models That Do Not Admit Restrictions (Infinite Models). Sum of Squares and Covariance Matrix Algorithms. Expected Mean Squares. Algorithm Applications. Least Squares Regression: Ordinary Least SquaresEstimation. Best Linear Unbiased Estimators. ANOVA Table for the Ordinary Least Squares Regression Function. Weighted Least Squares Regression. Lack of Fit Test. Partitioning the Sum of Squares Regression. The Model Y = X( + E in Complete, BalancedFactorials. Maximum Likelihood Estimation and Related Topics: Maximum Likelihood Estimators (MLEs) of ( and ( + 2. Invariance Property, Sufficiency and Completeness. ANOVA Methods for Finding Maximum Likelihood Estimators. The Likelihood Ratio Test for H( = h. Confidence Bands on Linear Combinations of (. Unbalanced Designs and Missing Data: Replication Matrices. Pattern Matrices and Missing Data. Using Replication and Pattern Matrices Together. Balanced Incomplete Block Designs: General Balanced Incomplete Block Design. Analysis of the General Case. Matrix Derivations of Kempthornes Inter- and Intra-Block Treatment Difference Estimators. Less Than Full Rank Models: Model Assumptions and Examples. The Mean Model Solution. Mean Model Analysis When cov(E) = ( + 2I - n. Estimable Functions. Mean Model Analysis When cov(E) = ( + 2V. The General Mixed Model: The Mixed Model Structure and Assumptions. Random Portion Analysis: Type I Sumof Squares Method. Random Portion Analysis: Restricted Maximum Likelihood Method. Random Portion Analysis: A Numerical Example. Fixed Portion Analysis. Fixed Portion Analysis: A Numerical Example. Appendixes. References. Subject Index.

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