Mathematical Statistics
Preview
Book Description
Traditional texts in mathematical statistics can seem - to some readers-heavily weighted with optimality theory of the various flavors developed in the 1940s and50s, and not particularly relevant to statistical practice. Mathematical Statistics stands apart from these treatments. While mathematically rigorous, its focus is on providing a set of useful tools that allow students to understand the theoretical underpinnings of statistical methodology.
The author concentrates on inferential procedures within the framework of parametric models, but - acknowledging that models are often incorrectly specified - he also views estimation from a non-parametric perspective. Overall, Mathematical Statistics places greater emphasis on frequentist methodology than on Bayesian, but claims no particular superiority for that approach. It does emphasize, however, the utility of statistical and mathematical software packages, and includes several sections addressing computational issues.
The result reaches beyond "nice" mathematics to provide a balanced, practical text that brings life and relevance to a subject so often perceived as irrelevant and dry.
Features
Table of Contents
INTRODUCTION TO PROBABILITY
Random Experiments
Probability Measures
Conditional Probability and Independence
Random Variables
Expected Values
RANDOM VECTORS AND JOINT DISTRIBUTIONS
Introduction
Discrete and Continuous Random Vectors
Conditional Distributions
Normal Distributions
Poisson Processes
Generating Random Variables
CONVERGENCE OF RANDOM VARIABLES
Introduction
Convergence in Probability and Distribution
WLLN
Proving Convergence in Distribution
CLT
Some Applications
Convergence with Probability 1
PRINCIPLES OF POINT ESTIMATION
Introduction
Statistical Models
Sufficiency
Point Estimation
Substitution Principle
Influence Curves
Standard Errors
Relative Efficiency
The Jackknife
LIKELIHOOD-BASED ESTIMATION
Introduction
The Likelihood Function
The Likelihood Principle
Asymptotics for MLEs
Misspecified Models
Nonparametric Maximum Likelihood Estimation
Numerical Computation
Bayesian Estimation
OPTIMAL ESTIMATION
Decision Theory
UMVUEs
The Cramér-Rao Lower Bound
Asymptotic Efficiency
INTERVAL ESTIMATION AND HYPOTHESIS TESTING
Confidence Intervals and Regions
Highest Posterior Density Regions
Hypothesis Testing
Likelihood Ratio Tests
Other Issues
LINEAR AND GENERALIZED LINEAR MODELS
Linear Models
Estimation
Testing
Non-Normal Errors
Generalized Linear Models
Quasi-Likelihood Models
GOODNESS OF FIT
Introduction
Tests Based on the Multinomial Distribution
Smooth Goodness of Fit Tests
REFERENCES
Each chapter also contains a Problems and Complements section
Reviews
Keith Knight's new book is a welcome addition to textbooks appropriate for masters-level theory courses. ... His is the best treatment of likelihood theory that I know at any level. ... I wish I had a nickel for every time I have been asked for recommended reading on likelihood theory and had to say one did not exist at this level. Now I can wholeheartedly recommend Mathematical Statistics.
C. GEYER, University of Minnesota in Journal of the American Statistical Association, June 2001
"…a very suitable text for teaching at an acceptable mathematical level…contains numerous examples and each chapter is followed by a rich choice of exercises…this makes the book excellent for teaching,"
-Short Book Reviews of the ISI
"well-written,,,far greater coverage of ides that are not standard in other mathematical statistics texts."
--M. S. Ridout, Institute of Mathematics and Statistics, University of Kent at Canterbury, UK in Biometrics
"…one of the five best textbooks on a beginning course on theoretical statistics providing a good grasp on the foundations of theoretical statistics. Primarily for graduate students with mathematical backgrounds in linear algebra, multivariable calculus, and some exposure to statistical methodology. Highly recommended for all academic libraries."
--D. V. Chopra, Wichita State University in CHOICE
"This books breaks away form more theoretically burdensome texts, focusing on providing a set of useful tools that help readers understand the theoretical under pinning of statistical methodology."
--SciTech Book News, March 2000
"This (hardback) book is one of the most up-to-date and easily understood texts in the field of mathematical statistics. The author has recognized the difficult nature of the subject and has done justice to the subject by finally producing one of the best well-rounded texts for graduate and senior undergraduate students. …well written and well structured. This text would be a very useful teaching tool.""
The Statistician, Vol. 50, Part 2, 2001