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Description:

This book builds theoretical statistics from the first principles of probability theory. Starting from the basics of probability, the authors develop the theory of statistical inference using techniques, definitions, and concepts that are statistical and are natural extensions and consequences of previous concepts. Intended for first-year graduate students, this book can be used for students majoring in statistics who have a solid mathematics background. It can also be used in a way that stresses the more practical uses of statistical theory, being more concerned with understanding basic statistical concepts and deriving reasonable statistical procedures for a variety of situations, and… More less concerned with formal optimality investigations.Less

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Book details

List price: $331.95 Edition: 2nd Copyright year: 2002 Publisher: Brooks/Cole Publication date: 6/18/2001 Binding: Hardcover Pages: 688 Size: 6.50" wide x 9.25" long x 1.25" tall Weight: 2.552 Language: English

AuthorTable of Contents

Probability Theory

Set Theory

Basics of Probability Theory

Axiomatic Foundations

The Calculus of Probabilities

Counting

Enumerating Outcomes

Conditional Probability and Independence

Random Variables

Distribution Functions

Density and Mass Functions

Exercises

Miscellanea

Transformations and Expectations

Distributions of Functions of a Random Variable

Expected Values

Moments and Moment Generating Functions

Differentiating Under an Integral Sign

Exercises

Miscellanea

Common Families of Distributions

Introduction

Discrete Distributions

Continuous Distributions

Exponential Families

Location and Scale Families

Inequalities and Identities

Probability Inequalities

Identities

Exercises

Miscellanea

Multiple Random Variables

Joint and Marginal Distributions

Conditional Distributions and Independence

Bivariate Transformations

Hierarchical Models and Mixture Distributions

Covariance and Correlation

Multivariate Distributions

Inequalities

Numerical Inequalities

Functional Inequalities

Exercises

Miscellanea

Properties of a Random Sample

Basic Concepts of Random Samples

Sums of Random Variables from a Random Sample

Sampling from the Normal Distribution

Properties of the Sample Mean and Variance

The Derived Distributions: Student's t and Snedecor's F

Order Statistics

Convergence Concepts

Convergence in Probability

Almost Sure Convergence

Convergence in Distribution

The Delta Method

Generating a Random Sample

Direct Methods

Indirect Methods

The Accept/Reject Algorithm

Exercises

Miscellanea

Principles of Data Reduction

Introduction

The Sufficiency Principle

Sufficient Statistics

Minimal Sufficient Statistics

Ancillary Statistics

Sufficient, Ancillary, and Complete Statistics

The Likelihood Principle

The Likelihood Function

The Formal Likelihood Principle

The Equivariance Principle

Exercises

Miscellanea

Point Estimation

Introduction

Methods of Finding Estimators

Method of Moments

Maximum Likelihood Estimators

Bayes Estimators

The EM Algorithm

Methods of Evaluating Estimators

Mean Squared Error

Best Unbiased Estimators

Sufficiency and Unbiasedness

Loss Function Optimality

Exercises

Miscellanea

Hypothesis Testing

Introduction

Methods of Finding Tests

Likelihood Ratio Tests

Bayesian Tests

Union-Intersection and Intersection-Union Tests

Methods of Evaluating Tests

Error Probabilities and the Power Function

Most Powerful Tests

Sizes of Union-Intersection and Intersection-Union Tests

p-Values

Loss Function Optimality

Exercises

Miscellanea

Interval Estimation

Introduction

Methods of Finding Interval Estimators

Inverting a Test Statistic

Pivotal Quantities

Pivoting the CDF

Bayesian Intervals

Methods of Evaluating Interval Estimators

Size and Coverage Probability

Test-Related Optimality

Bayesian Optimality

Loss Function Optimality

Exercises

Miscellanea

Asymptotic Evaluations

Point Estimation

Consistency

Efficiency

Calculations and Comparisons

Bootstrap Standard Errors

Robustness

The Mean and the Median

M-Estimators

Hypothesis Testing

Asymptotic Distribution of LRTs

Other Large-Sample Tests

Interval Estimation

Approximate Maximum Likelihood Intervals

Other Large-Sample Intervals

Exercises

Miscellanea

Analysis of Variance and Regression

Introduction

Oneway Analysis of Variance

Model and Distribution Assumptions

The Classic ANOVA Hypothesis

Inferences Regarding Linear Combinations of Means

The ANOVA F Test

Simultaneous Estimation of Contrasts

Partitioning Sums of Squares

Simple Linear Regression

Least Squares: A Mathematical Solution

Best Linear Unbiased Estimators: A Statistical Solution

Models and Distribution Assumptions

Estimation and Testing with Normal Errors

Estimation and Prediction at a Specified x = x[subscript 0]

Simultaneous Estimation and Confidence Bands

Exercises

Miscellanea

Regression Models

Introduction

Regression with Errors in Variables

Functional and Structural Relationships

A Least Squares Solution

Maximum Likelihood Estimation

Confidence Sets

Logistic Regression

The Model

Estimation

Robust Regression

Exercises

Miscellanea

Computer Algebra

Table of Common Distributions

References

Author Index

Subject Index

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