Elsevier S & T

A Course in Mathematical Statistics

Roussas, George G.

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Table of contents

Cover
Contentsvii
Preface to the Second Editionxv
Preface to the First Editionxviii
Chapter 1. Basic Concepts of Set Theory1
1.1 Some Definitions and Notation1
1.2* Fields and σ-Fields8
Chapter 2. Some Probabilistic Concepts and Results14
2.1 Probability Functions and Some Basic Properties and Results14
2.2 Conditional Probability21
2.3 Independence27
2.4 Combinatorial Results34
2.5* Product Probability Spaces45
2.6* The Probability of Matchings47
Chapter 3. On Random Variables and Their Distributions53
3.1 Some General Concepts53
3.2 Discrete Random Variables (and Random Vectors)55
3.3 Continuous Random Variables (and Random Vectors)65
3.4 The Poisson Distribution as an Approximation to the Binomial Distribution and the Binomial Distr79
3.5* Random Variables as Measurable Functions and Related Results82
Chapter 4. Distribution Functions, Probability Densities, and Their Relationship85
4.1 The Cumulative Distribution Function (c.d.f. or d.f.) of a Random Vector„Basic Properties of t85
4.2 The d.f. of a Random Vector and Its Properties„Marginal and Conditional d.f.s and p.d.f.s91
4.3 Quantiles and Modes of a Distribution99
4.4* Justification of Statements 1 and 2102
Chapter 5. Moments of Random Variables„Some Moment and Probability Inequalities106
5.1 Moments of Random Variables106
5.2 Expectations and Variances of Some r.v.’s114
5.3 Conditional Moments of Random Variables122
5.4 Some Important Applications: Probability and Moment Inequalities125
5.5 Covariance, Correlation Coefficient and Its Interpretation129
5.6* Justification of Relation (2) in Chapter 2134
Chapter 6. Characteristic Functions, Moment Generating Functions and Related Theorems138
6.1 Preliminaries138
6.2 Definitions and Basic Theorems„The One-Dimensional Case140
6.3 The Characteristic Functions of Some Random Variables146
6.4 Definitions and Basic Theorems„The Multidimensional Case150
6.5 The Moment Generating Function and Factorial Moment Generating Function of a Random Variable153
Chapter 7. Stochastic Independence with Some Applications164
7.1 Stochastic Independence: Criteria of Independence164
7.2 Proof of Lemma 2 and Related Results170
7.3 Some Consequences of Independence173
7.4* Independence of Classes of Events and Related Results177
Chapter 8. Basic Limit Theorems180
8.1 Some Modes of Convergence180
8.2 Relationships Among the Various Modes of Convergence182
8.3 The Central Limit Theorem187
8.4 Laws of Large Numbers196
8.5 Further Limit Theorems199
8.6* Pólya’s Lemma and Alternative Proof of the WLLN206
Chapter 9. Transformations of Random Variables and Random Vectors212
9.1 The Univariate Case212
9.2 The Multivariate Case219
9.3 Linear Transformations of Random Vectors235
9.4 The Probability Integral Transform242
Chapter 10. Order Statistics and Related Theorems245
10.1 Order Statistics and Related Distributions245
10.2 Further Distribution Theory: Probability of Coverage of a Population Quantile256
Chapter 11. Sufficiency and Related Theorems259
11.1 Sufficiency: Definition and Some Basic Results260
11.2 Completeness271
11.3 Unbiasedness„Uniqueness274
11.4 The Exponential Family of p.d.f.’s. One-Dimensional Parameter Case276
11.5 Some Multiparameter Generalizations281
Chapter 12. Point Estimation284
12.1 Introduction284
12.2 Criteria for Selecting an Estimator: Unbiasedness, Minimum Variance285
12.3 The Case of Availability of Complete Sufficient Statistics287
12.4 The Case Where Complete Sufficient Statistics Are Not Available or May Not Exist: Cramér-Rao I293
12.5 Criteria for Selecting an Estimator: The Maximum Likelihood Principle302
12.6 Criteria for Selecting an Estimator: The Decision- Theoretic Approach309
12.7 Finding Bayes Estimators312
12.8 Finding Minimax Estimators318
12.9 Other Methods of Estimation320
12.10 Asymptotically Optimal Properties of Estimators322
12.11 Closing Remarks325
Chapter 13. Testing Hypotheses327
13.1 General Concepts of the Neyman-Pearson Testing Hypotheses Theory327
13.2 Testing a Simple Hypothesis Against a Simple Alternative329
13.3 UMP Tests for Testing Certain Composite Hypotheses337
13.4 UMPU Tests for Testing Certain Composite Hypotheses349
13.5 Testing the Parameters of a Normal Distribution353
13.6 Comparing the Parameters of Two Normal Distributions357
13.7 Likelihood Ratio Tests361
13.8 Applications of LR Tests: Contingency Tables, Goodness-of-Fit Tests370
13.9 Decision-Theoretic Viewpoint of Testing Hypotheses375
Chapter 14. Sequential Procedures382
14.1 Some Basic Theorems of Sequential Sampling382
14.2 Sequential Probability Ratio Test388
14.3 Optimality of the SPRT-Expected Sample Size393
14.4 Some Examples394
Chapter 15. Confidence Regions„Tolerance Intervals397
15.1 Confidence Intervals397
15.2 Some Examples398
15.3 Confidence Intervals in the Presence of Nuisance Parameters407
15.4 Confidence Regions„Approximate Confidence Intervals410
15.5 Tolerance Intervals413
Chapter 16. The General Linear Hypothesis416
16.1 Introduction of the Model416
16.2 Least Square Estimators„Normal Equations418
16.3 Canonical Reduction of the Linear Model—Estimation of σ424
16.4 Testing Hypotheses About η= E(Y)429
16.5 Derivation of the Distribution of the F Statistic433
Chapter 17. Analysis of Variance440
17.1 One-way Layout (or One-way Classi.cation) with the Same Number of Observations Per Cell440
17.2 Two-way Layout (Classification) with One Observation Per Cell446
17.3 Two-way Layout (Classification) with K (> 2) Observations Per Cell452
17.4 A Multicomparison method458
Chapter 18. The Multivariate Normal Distribution463
18.1 Introduction463
18.2 Some Properties of Multivariate Normal Distributions467
18.3 Estimation of μ and Σ and a Test of Independence469
Chapter 19. Quadratic Forms476
19.1 Introduction476
19.2 Some Theorems on Quadratic Forms477
Chapter 20. Nonparametric Inference485
20.1 Nonparametric Estimation485
20.2 Nonparametric Estimation of a p.d.f.487
20.3 Some Nonparametric Tests490
20.4 More About Nonparametric Tests: Rank Tests493
20.5 Sign Test496
20.6 Relative Asymptotic Efficiency of Tests497
Appendix I. Topics from Vector and Matrix Algebra499
I.1 Basic Definitions in Vector Spaces499
I.2 Some Theorems on Vector Spaces501
I.3 Basic Definitions About Matrices502
I.4 Some Theorems About Matrices and Quadratic Forms504
Appendix II. Noncentral t, X2 and F-Distributions508
II.1 Noncentral t-Distribution508
II.2 Noncentral X2-Distribution508
II.3 Noncentral F-Distribution509
Appendix III. Tables511
1 The Cumulative Binomial Distribution511
2 The Cumulative Poisson Distribution520
3 The Normal Distribution523
4 Critical Values for Student’s t-Distribution526
5 Critical Values for the Chi-Square Distribution529
6 Critical Values for the F-Distribution532
7 Table of Selected Discrete and Continuous Distributions and Some of Their Characteristics542
Some Notation and Abbreviations545
Answers to Selected Exercises547
Index561

Book details

Vendor Elsevier S & T
SKU 9780125993159
ISBN-13 9780080493145
Author Roussas, George G.
Edition 2nd
Category Mathematics
Subject Mathematical Analysis

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A Course in Mathematical Statistics, Second Edition, contains enough material for a year-long course in probability and statistics for advanced undergraduate or first-year graduate students, or it can be used independently for a one-semester (or even one-quarter) course in probability alone. It bridges the gap between high and intermediate level texts so students without a sophisticated mathematical background can assimilate a fairly broad spectrum of the theorems and results from mathematical statistics. The coverage is extensive, and consists of probability and distribution theory, and statistical inference.

* Contains 25% new material
* Includes the most complete coverage of sufficiency
* Transformation of Random Vectors
* Sufficiency / Completeness / Exponential Families
* Order Statistics
* Elements of Nonparametric Density Estimation
* Analysis of Variance (ANOVA)
* Regression Analysis
* Linear Models

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