Introduction to Mathematical Statistics (What's New in Statistics)

Front Cover
Title Page
Copyright Page
Dedication Page
Contents
Preface
1 Probability and Distributions
	1.1 Introduction
	1.2 Sets
		1.2.1 Review of Set Theory
		1.2.2 Set Functions
	1.3 The Probability Set Function
		1.3.1 Counting Rules
		1.3.2 Additional Properties of Probability
	1.4 Conditional Probability and Independence
		1.4.1 Independence
		1.4.2 Simulations
	1.5 Random Variables
	1.6 Discrete Random Variables
		1.6.1 Transformations
	1.7 Continuous Random Variables
		1.7.1 Quantiles
		1.7.2 Transformations
		1.7.3 Mixtures of Discrete and Continuous Type Distributions
	1.8 Expectation of a Random Variable
		1.8.1 R Computation for an Estimation of the Expected Gain
	1.9 Some Special Expectations
	1.10 Important Inequalities
2 Multivariate Distributions
	2.1 Distributions of Two Random Variables
		2.1.1 Marginal Distributions
		2.1.2 Expectation
	2.2 Transformations: Bivariate Random Variables
	2.3 Conditional Distributions and Expectations
	2.4 Independent Random Variables
	2.5 The Correlation Coefficient
	2.6 Extension to Several Random Variables
		2.6.1 *Multivariate Variance-Covariance Matrix
	2.7 Transformations for Several Random Variables
	2.8 Linear Combinations of Random Variables
3 Some Special Distributions
	3.1 The Binomial and Related Distributions
		3.1.1 Negative Binomial and Geometric Distributions
		3.1.2 Multinomial Distribution
		3.1.3 Hypergeometric Distribution
	3.2 The Poisson Distribution
	3.3 The Γ, χ2, and β Distributions
		3.3.1 The χ2-Distribution
		3.3.2 The β-Distribution
	3.4 The Normal Distribution
		3.4.1 *Contaminated Normals
	3.5 The Multivariate Normal Distribution
		3.5.1 Bivariate Normal Distribution
		3.5.2 *Multivariate Normal Distribution, General Case
		3.5.3 *Applications
	t- and F-Distributions
		3.6.1 The t-distribution
		3.6.2 The F-distribution
		3.6.3 Student’s Theorem
	3.7 *Mixture Distributions
4 Some Elementary Statistical Inferences
	4.1 Sampling and Statistics
		4.1.1 Point Estimators
		4.1.2 Histogram Estimates of Pmfs and Pdfs
	4.2 Confidence Intervals
		4.2.1 Confidence Intervals for Difference in Means
		4.2.2 Confidence Interval for Difference in Proportions
	4.3 *Confidence Intervals for Parameters of Discrete Distributions
	4.4 Order Statistics
		4.4.1 Quantiles
		4.4.2 Confidence Intervals for Quantiles
	4.5 Introduction to Hypothesis Testing
	4.6 Additional Comments About Statistical Tests
		4.6.1 Observed Significance Level, p-value
	4.7 Chi-Square Tests
	4.8 The Method of Monte Carlo
		4.8.1 Accept–Reject Generation Algorithm
	4.9 Bootstrap Procedures
		4.9.1 Percentile Bootstrap Confidence Intervals
		4.9.2 Bootstrap Testing Procedures
	4.10 *Tolerance Limits for Distributions
5 Consistency and Limiting Distributions
	5.1 Convergence in Probability
		5.1.1 Sampling and Statistics
	5.2 Convergence in Distribution
		5.2.1 Bounded in Probability
		5.2.2 Δ-Method
		5.2.3 Moment Generating Function Technique
	5.3 Central Limit Theorem
	5.4 *Extensions to Multivariate Distributions
6 Maximum Likelihood Methods
	6.1 Maximum Likelihood Estimation
	6.2 Rao–Cram´er Lower Bound and Efficiency
	6.3 Maximum Likelihood Tests
	6.4 Multiparameter Case: Estimation
	6.5 Multiparameter Case: Testing
	6.6 The EM Algorithm
7 Sufficiency
	7.1 Measures of Quality of Estimators
	7.2 A Sufficient Statistic for a Parameter
	7.3 Properties of a Sufficient Statistic
	7.4 Completeness and Uniqueness
	7.5 The Exponential Class of Distributions
	7.6 Functions of a Parameter
		7.6.1 Bootstrap Standard Errors
	7.7 The Case of Several Parameters
	7.8 Minimal Sufficiency and Ancillary Statistics
	7.9 Sufficiency, Completeness, and Independence
8 Optimal Tests of Hypotheses
	8.1 Most Powerful Tests
	8.2 Uniformly Most Powerful Tests
	8.3 Likelihood Ratio Tests
		8.3.1 Likelihood Ratio Tests for Testing Means of Normal Distributions
		8.3.2 Likelihood Ratio Tests for Testing Variances of Normal Distributions
	8.4 *The Sequential Probability Ratio Test
	8.5 *Minimax and Classification Procedures
		8.5.1 Minimax Procedures
		8.5.2 Classification
9 Inferences About Normal Linear Models
	9.1 Introduction
	9.2 One-Way ANOVA
	9.3 Noncentral χ2 and F-Distributions
	9.4 Multiple Comparisons
	9.5 Two-Way ANOVA
		9.5.1 Interaction Between Factors
	9.6 A Regression Problem
		9.6.1 Maximum Likelihood Estimates
		9.6.2 *Geometry of the Least Squares Fit
	9.7 A Test of Independence
	9.8 The Distributions of Certain Quadratic Forms
	9.9 The Independence of Certain Quadratic Forms
10 Nonparametric and Robust Statistics
	10.1 Location Models
	10.2 Sample Median and the Sign Test
		10.2.1 Asymptotic Relative Efficiency
		10.2.2 Estimating Equations Based on the Sign Test
		10.2.3 Confidence Interval for the Median
	10.3 Signed-Rank Wilcoxon
		10.3.1 Asymptotic Relative Efficiency
		10.3.2 Estimating Equations Based on Signed-Rank Wilcoxon
		10.3.3 Confidence Interval for the Median
		10.3.4 Monte Carlo Investigation
	10.4 Mann–Whitney–Wilcoxon Procedure
		10.4.1 Asymptotic Relative Efficiency
		10.4.2 Estimating Equations Based on the Mann–Whitney– Wilcoxon
		10.4.3 Confidence Interval for the Shift Parameter Δ
		10.4.4 Monte Carlo Investigation of Power
	10.5 *General Rank Scores
		10.5.1 Efficacy
		10.5.2 Estimating Equations Based on General Scores
		10.5.3 Optimization: Best Estimates
	10.6 *Adaptive Procedures
	10.7 Simple Linear Model
	10.8 Measures of Association
		10.8.1 Kendall’s τ
		10.8.2 Spearman’s Rho
	10.9 Robust Concepts
		10.9.1 Location Model
		10.9.2 Linear Model
11 Bayesian Statistics
	11.1 Bayesian Procedures
		11.1.1 Prior and Posterior Distributions
		11.1.2 Bayesian Point Estimation
		11.1.3 Bayesian Interval Estimation
		11.1.4 Bayesian Testing Procedures
		11.1.5 Bayesian Sequential Procedures
	11.2 More Bayesian Terminology and Ideas
	11.3 Gibbs Sampler
	11.4 Modern Bayesian Methods
		11.4.1 Empirical Bayes
A Mathematical Comments
	A.1 Regularity Conditions
	A.2 Sequences
B R Primer
	B.1 Basics
	B.2 Probability Distributions
	B.3 R Functions
	B.4 Loops
	B.5 Input and Output
	B.6 Packages
C Lists of Common Distributions
D Tables of Distributions
E References
F Answers to Selected Exercises
Index
Back Cover