A Course in Mathematical Statistics
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Contents
Preface to the Second Edition
Preface to the First Edition
Chapter 1. Basic Concepts of Set Theory
Chapter 2. Some Probabilistic Concepts and Results
2.1 Probability Functions and Some Basic Properties and Results
2.2 Conditional Probability
2.3 Independence
2.4 Combinatorial Results
2.5* Product Probability Spaces
2.6* The Probability of Matchings
Chapter 3. On Random Variables and Their Distributions
3.1 Some General Concepts
3.2 Discrete Random Variables (and Random Vectors)
3.3 Continuous Random Variables (and Random Vectors)
3.4 The Poisson Distribution as an Approximation to the Binomial Distribution and the Binomial Distribution as an Approximation to the Hypergeometric Distribution
3.5* Random Variables as Measurable Functions and Related Results
Chapter 4. Distribution Functions, Probability Densities, and Their Relationship
4.1 The Cumulative Distribution Function (c.d.f. or d.f.) of a Random Vector—Basic Properties of the d.f. of a Random Variable
4.2 The d.f. of a Random Vector and Its Properties—Marginal and Conditional d.f.’s and p.d.f.’s
4.3 Quantiles and Modes of a Distribution
4.4* Justification of Statements 1 and 2
Chapter 5. Moments of Random Variables—Some Moment and Probability Inequalities
5.1 Moments of Random Variables
5.2 Expectations and Variances of Some r.v.’s
5.3 Conditional Moments of Random Variables
5.4 Some Important Applications: Probability and Moment Inequalities
5.5 Covariance, Correlation Coefficient and Its Interpretation
5.6* Justification of Relation (2) in Chapter 2
Chapter 6. Characteristic Functions, Moment Generating Functions and Related Theorems
6.1 Preliminaries
6.2 Definitions and Basic Theorems—The One-Dimensional Case
6.3 The Characteristic Functions of Some Random Variables
6.4 Definitions and Basic Theorems—The Multidimensional Case
6.5 The Moment Generating Function and Factorial Moment Generating Function of a Random Variable
Chapter 7. Stochastic Independence with Some Applications
7.1 Stochastic Independence: Criteria of Independence
7.2 Proof of Lemma 2 and Related Results
7.3 Some Consequences of Independence
7.4* Independence of Classes of Events and Related Results
Chapter 8. Basic Limit Theorems
8.1 Some Modes of Convergence
8.2 Relationships Among the Various Modes of Convergence
8.3 The Central Limit Theorem
8.4 Laws of Large Numbers
8.5 Further Limit Theorems
8.6* Pólya’s Lemma and Alternative Proof of the WLLN
Chapter 9. Transformations of Random Variables and Random Vectors
9.1 The Univariate Case
9.2 The Multivariate Case
9.3 Linear Transformations of Random Vectors
9.4 The Probability Integral Transform
Chapter 10. Order Statistics and Related Theorems
Chapter 11. Sufficiency and Related Theorems
11.1 Sufficiency: Definition and Some Basic Results
11.2 Completeness
11.3 Unbiasedness—Uniqueness
11.4 The Exponential Family of p.d.f.’s. One-Dimensional Parameter Case
11.5 Some Multiparameter Generalizations
Chapter 12. Point Estimation
12.1 Introduction
12.2 Criteria for Selecting an Estimator: Unbiasedness, Minimum Variance
12.3 The Case of Availability of Complete Sufficient Statistics
12.4 The Case Where Complete Sufficient Statistics Are Not Available or May Not Exist: Cramér-Rao Inequality
12.5 Criteria for Selecting an Estimator: The Maximum Likelihood Principle
12.6 Criteria for Selecting an Estimator: The Decision-Theoretic Approach
12.7 Finding Bayes Estimators
12.8 Finding Minimax Estimators
12.9 Other Methods of Estimation
12.10 Asymptotically Optimal Properties of Estimators
12.11 Closing Remarks
Chapter 13. Testing Hypotheses
13.1 General Concepts of the Neyman–Pearson Testing Hypotheses Theory
13.2 Testing a Simple Hypothesis Against a Simple Alternative
13.3 UMP Tests for Testing Certain Composite Hypotheses
13.4 UMPU Tests for Testing Certain Composite Hypotheses
13.5 Testing the Parameters of a Normal Distribution
13.6 Comparing the Parameters of Two Normal Distributions
13.7 Likelihood Ratio Tests
13.8 Applications of LR Tests: Contingency Tables, Goodness-of-Fit Tests
13.9 Decision-Theoretic Viewpoint of Testing Hypotheses
Chapter 14. Sequential Procedures
14.1 Some Basic Theorems of Sequential Sampling
14.2 Sequential Probability Ratio Test
14.3 Optimality of the SPRT-Expected Sample Size
14.4 Some Examples
Chapter 15. Confidence Regions—Tolerance Intervals
15.1 Confidence Intervals
15.2 Some Examples
15.3 Confidence Intervals in the Presence of Nuisance Parameters
15.4 Confidence Regions—Approximate Confidence Intervals
15.5 Tolerance Intervals
Chapter 16. The General Linear Hypothesis
16.1 Introduction of the Model
16.2 Least Square Estimators—Normal Equations
16.3 Canonical Reduction of the Linear Model—Estimation of σ
16.4 Testing Hypotheses About η= E(Y)
16.5 Derivation of the Distribution of the F Statistic
Chapter 17. Analysis of Variance
17.1 One-way Layout (or One-way Classification) with the Same Number of Observations Per Cell
17.2 Two-way Layout (Classification) with One Observation Per Cell
17.3 Two-way Layout (Classification) with K (> 2) Observations Per Cell
17.4 A Multicomparison method
Chapter 18. The Multivariate Normal Distribution
Chapter 19. Quadratic Forms
Chapter 20. Nonparametric Inference
20.1 Nonparametric Estimation
20.2 Nonparametric Estimation of a p.d.f.
20.3 Some Nonparametric Tests
20.4 More About Nonparametric Tests: Rank Tests
20.5 Sign Test
20.6 Relative Asymptotic Efficiency of Tests
Appendix I. Topics from Vector and Matrix Algebra
I.1 Basic Definitions in Vector Spaces
I.2 Some Theorems on Vector Spaces
I.3 Basic Definitions About Matrices
I.4 Some Theorems About Matrices and Quadratic Forms
Appendix II. Noncentral t, X2 and F-Distributions
II.1 Noncentral t-Distribution
II.2 Noncentral X2-Distribution
II.3 Noncentral F-Distribution
Appendix III. Tables
1 The Cumulative Binomial Distribution
2 The Cumulative Poisson Distribution
3 The Normal Distribution
4 Critical Values for Student’s t-Distribution
5 Critical Values for the Chi-Square Distribution
6 Critical Values for the F-Distribution
7 Table of Selected Discrete and Continuous Distributions and Some of Their Characteristics
Some Notation and Abbreviations
Answers to Selected Exercises
Index