This gracefully organized text reveals the rigorous theory of probability and statistical inference in the style of a tutorial, using worked examples, exercises, figures, tables, and computer simulations to develop and illustrate concepts. Drills and boxed summaries emphasize and reinforce important ideas and special techniques. Beginning with a review of the basic concepts and methods in probability theory, moments, and moment generating functions, the author moves to more intricate topics. "Introductory Statistical Inference" studies multivariate random variables, exponential families of distributions, and standard probability inequalities. It develops the Helmert transformation for normal distributions, introduces the notions of convergence, and spotlights the central limit theorems. This book highlights sampling distributions, Basu's theorem, Rao-Blackwellization and the Cramr-Rao inequality. The text also provides in-depth coverage of Lehmann-Scheff theorems, focusing on tests of hypotheses, it describes Bayesian methods and the Bayes' estimator, and develops large-sample inference. The author provides a historical context for statistics and statistical discoveries and answers to a majority of the end-of-chapter exercises. Designed primarily for a one-semester, first-year graduate course in probability and statistical inference, this text serves readers from varied backgrounds, ranging from engineering, economics, agriculture, and bioscience to finance, financial mathematics, operations and information management, and psychology.
Probability and Distributions
Introduction
About Sets
Axiomatic Development of Probability
Conditional Probability and Independent Events
Discrete Random Variables
Continuous Random Variables
Some Useful Distributions
Exercises and Complements
Moments and Generating Functions
Introduction
Expectation and Variance
Moments and Moment Generating Function
Determination of a Distribution via MGF
Probability Generating Function
Exercises and Complements
Multivariate Random Variables
Introduction
Probability Distributions
Covariances and Correlation Coefficient
Independence of Random Variables
Bivariate Normal Distribution
Correlation Coefficient and Independence
Exponential Family
Selected Probability Inequalities
Exercises and Complements
Sampling Distribution
Introduction
Moment Generating Function Approach
Order Statistics
Transformation
Special Sampling Distributions
Multivariate Normal Distribution
Selected Reviews in Matrices
Exercises and Complements
Notions of Convergence
Introduction
Convergence in Probability
Convergence in Distribution
Convergence of Chi-Square, t, and F distributions
Exercises and Complements
Sufficiency, Completeness, and Ancillarity
Introduction
Sufficiency
Minimal Sufficiency
Information
Ancillarity
Completeness
Exercises and Complements
Point Estimation
Introduction
Maximum Likelihood Estimator
Criteria to Compare Estimators
Improved Unbiased Estimators via Sufficiency
Uniformly Minimum Variance Unbiased Estimator
Consistent Estimator
Exercises and Complements
Tests of Hypotheses
Introduction
Error Probabilities and Power Function
Simple Null vs. Simple Alternative
One-Sided Composite Alternative
Simple Null vs. Two-Sided Alternative
Exercises and Complements
Confidence Intervals
Introduction
One-Sample Problems
Two-Sample Problems
Exercises and Complements
Bayesian Methods
Introduction
Prior and Posterior Distributions
Conjugate Prior
Point Estimation
Examples with a Nonconjugate Prior
Exercises and Complements
Likelihood Ratio and Other Tests
Introduction
One-Sample LR Tests: Normal
Two-Sample LR Tests: Independent Normal
Bivariate Normal
Exercises and Complements
Large-Sample Methods
Introduction
Maximum Likelihood Estimation
Asymptotic Relative Efficiency
Confidence Intervals and Tests of Hypotheses
Variance Stabilizing Transformation
Exercises and Complements
Abbreviations, Historical Notes, and Tables
Abbreviations and Notations
Historical Notes
Selected Statistical Tables
References
Answers: Selected Exercises
Author Index
Subject Index
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