
Examples and Problems in Mathematical Statistics
[Book Description]
Provides the necessary skills to solve problems in mathematical statistics through theory, concrete examples, and exercises With a clear and detailed approach to the fundamentals of statistical theory, Examples and Problems in Mathematical Statistics uniquely bridges the gap between theory andapplication and presents numerous problem-solving examples that illustrate the relatednotations and proven results. Written by an established authority in probability and mathematical statistics, each chapter begins with a theoretical presentation to introduce both the topic and the important results in an effort to aid in overall comprehension. Examples are then provided, followed by problems, and finally, solutions to some of the earlier problems.In addition, Examples and Problems in Mathematical Statistics features: * Over 160 practical and interesting real-world examples from a variety of fields including engineering, mathematics, and statistics to help readers become proficient in theoretical problem solving * More than 430 unique exercises with select solutions * Key statistical inference topics, such as probability theory, statistical distributions, sufficient statistics, information in samples, testing statistical hypotheses, statistical estimation, confidence and tolerance intervals, large sample theory, and Bayesian analysis Recommended for graduate-level courses in probability and statistical inference, Examples and Problems in Mathematical Statistics is also an ideal reference for applied statisticians and researchers.
[Table of Contents]
Preface xv
List of Random Variables xvii
List of Abbreviations xix
1 Basic Probability Theory
Part I Theory 1 1 (1)
1.1 Operations on Sets 1 (1)
1.2 Algebra and σ-Fields 2 (2)
1.3 Probability Spaces 4 (2)
1.4 Conditional Probabilities and 6 (2)
Independence
1.5 Random Variables and Their 8 (4)
Distributions
1.6 The Lebesgue and Stieltjes Integrals 12 (9)
1.6.1 General Definition of Expected 12 (5)
Value: The Lebesgue Integral
1.6.2 The Stieltjes-Riemann Integral 17 (2)
1.6.3 Mixtures of Discrete and 19 (1)
Absolutely Continuous Distributions
1.6.4 Quantiles of Distributions 19 (1)
1.6.5 Transformations 20 (1)
1.7 Joint Distributions, Conditional 21 (5)
Distributions and Independence
1.7.1 Joint Distributions 21 (2)
1.7.2 Conditional Expectations: General 23 (3)
Definition
1.7.3 Independence 26 (1)
1.8 Moments and Related Functionals 26 (9)
1.9 Modes of Convergence 35 (4)
1.10 Weak Convergence 39 (2)
1.11 Laws of Large Numbers 41 (3)
1.11.1 The Weak Law of Large Numbers 41 (1)
(WLLN)
1.11.2 The Strong Law of Large Numbers 42 (2)
(SLLN)
1.12 Central Limit Theorem 44 (3)
1.13 Miscellaneous Results 47 (59)
1.13.1 Law of the Iterated Logarithm 48 (1)
1.13.2 Uniform Integrability 48 (4)
1.13.3 Inequalities 52 (1)
1.13.4 The Delta Method 53 (2)
1.13.5 The Symbols op and Op 55 (1)
1.13.6 The Empirical Distribution and 55 (1)
Sample Quantiles
Part II Examples 56 (17)
Part III Problems 73 (20)
Part IV Solutions To Selected Problems 93 (13)
2 Statistical Distributions 106 (85)
Part I Theory 106 (1)
2.1 Introductory Remarks 106 (1)
2.2 Families of Discrete Distributions 106 (3)
2.2.1 Binomial Distributions 106 (1)
2.2.2 Hypergeometric Distributions 107 (1)
2.2.3 Poisson Distributions 108 (1)
2.2.4 Geometric, Pascal, and Negative 108 (1)
Binomial Distributions
2.3 Some Families of Continuous 109 (9)
Distributions
2.3.1 Rectangular Distributions 109 (2)
2.3.2 Beta Distributions 111 (1)
2.3.3 Gamma Distributions 111 (1)
2.3.4 Weibull and Extreme Value 112 (1)
Distributions
2.3.5 Normal Distributions 113 (1)
2.3.6 Normal Approximations 114 (4)
2.4 Transformations 118 (2)
2.4.1 One-to-One Transformations of 118 (1)
Several Variables
2.4.2 Distribution of Sums 118 (1)
2.4.3 Distribution of Ratios 118 (2)
2.5 Variances and Covariances of Sample 120 (2)
Moments
2.6 Discrete Multivariate Distributions 122 (3)
2.6.1 The Multinomial Distribution 122 (1)
2.6.2 Multivariate Negative Binomial 123 (1)
2.6.3 Multivariate Hypergeometric 124 (1)
Distributions
2.7 Multinomial Distributions 125 (5)
2.7.1 Basic Theory 125 (2)
2.7.2 Distribution of Subvectors and 127 (2)
Distributions of Linear Forms
2.7.3 Independence of Linear Forms 129 (1)
2.8 Distributions of Symmetric Quadratic 130 (2)
Forms of Normal Variables
2.9 Independence of Linear and Quadratic 132 (1)
Forms of Normal Variables
2.10 The Order Statistics 133 (2)
2.11 t-Distributions 135 (3)
2.12 F-Distributions 138 (4)
2.13 The Distribution of the Sample 142 (2)
Correlation
2.14 Exponential Type Families 144 (2)
2.15 Approximating the Distribution of 146 (45)
the Sample Mean: Edgeworth and
Saddlepoint Approximations
2.15.1 Edgeworth Expansion 147 (2)
2.15.2 Saddlepoint Approximation 149 (1)
Part II Examples 150 (17)
Part III Problems 167 (14)
Part IV Solutions To Selected Problems 181 (10)
3 Sufficient Statistics and the Information 191 (55)
in Samples
Part I Theory 191 (1)
3.1 Introduction 191 (1)
3.2 Definition and Characterization of 192 (8)
Sufficient Statistics
3.2.1 Introductory Discussion 192 (2)
3.2.2 Theoretical Formulation 194 (6)
3.3 Likelihood Functions and Minimal 200 (2)
Sufficient Statistics
3.4 Sufficient Statistics and Exponential 202 (1)
Type Families
3.5 Sufficiency and Completeness 203 (2)
3.6 Sufficiency and Ancillarity 205 (1)
3.7 Information Functions and Sufficiency 206 (6)
3.7.1 The Fisher Information 206 (4)
3.7.2 The Kullback-Leibler Information 210 (2)
3.8 The Fisher Information Matrix 212 (2)
3.9 Sensitivity to Changes in Parameters 214 (32)
3.9.1 The Hellinger Distance 214 (2)
Part II Examples 216 (14)
Part III Problems 230 (6)
Part IV Solutions To Selected Problems 236 (10)
4 Testing Statistical Hypotheses 246 (75)
Part I Theory 246 (1)
4.1 The General Framework 246 (2)
4.2 The Neyman--Pearson Fundamental Lemma 248 (3)
4.3 Testing One-Sided Composite 251 (3)
Hypotheses in MLR Models
4.4 Testing Two-Sided Hypotheses in 254 (2)
One-Parameter Exponential Families
4.5 Testing Composite Hypotheses with 256 (4)
Nuisance Parameters---Unbiased Tests
4.6 Likelihood Ratio Tests 260 (11)
4.6.1 Testing in Normal Regression 261 (4)
Theory
4.6.2 Comparison of Normal Means: The 265 (6)
Analysis of Variance
4.7 The Analysis of Contingency Tables 271 (4)
4.7.1 The Structure of Multi-Way 271 (1)
Contingency Tables and the Statistical
Model
4.7.2 Testing the Significance of 271 (2)
Association
4.7.3 The Analysis of 2 x 2 Tables 273 (1)
4.7.4 Likelihood Ratio Tests for 274 (1)
Categorical Data
4.8 Sequential Testing of Hypotheses 275 (46)
4.8.1 The Wald Sequential Probability 276 (7)
Ratio Test
Part II Examples 283 (15)
Part III Problems 298 (9)
Part IV Solutions To Selected Problems 307 (14)
5 Statistical Estimation 321 (85)
Part I Theory 321 (1)
5.1 General Discussion 321 (1)
5.2 Unbiased Estimators 322 (6)
5.2.1 General Definition and Example 322 (1)
5.2.2 Minimum Variance Unbiased 322 (1)
Estimators
5.2.3 The Cramer-Rao Lower Bound for 323 (3)
the One-Parameter Case
5.2.4 Extension of the Cramer--Rao 326 (1)
Inequality to Multiparameter Cases
5.2.5 General Inequalities of the 327 (1)
Cramer--Rao Type
5.3 The Efficiency of Unbiased Estimators 328 (3)
in Regular Cases
5.4 Best Linear Unbiased and 331 (4)
Least-Squares Estimators
5.4.1 BLUEs of the Mean 331 (1)
5.4.2 Least-Squares and BLUEs in Linear 332 (2)
Models
5.4.3 Best Linear Combinations of Order 334 (1)
Statistics
5.5 Stabilizing the LSE: Ridge Regressions 335 (2)
5.6 Maximum Likelihood Estimators 337 (4)
5.6.1 Definition and Examples 337 (1)
5.6.2 MLEs in Exponential Type Families 338 (1)
5.6.3 The Invariance Principle 338 (1)
5.6.4 MLE of the Parameters of 339 (2)
Tolerance Distributions
5.7 Equivariant Estimators 341 (5)
5.7.1 The Structure of Equivariant 341 (2)
Estimators
5.7.2 Minimum MSE Equivariant Estimators 343 (1)
5.7.3 Minimum Risk Equivariant 343 (1)
Estimators
5.7.4 The Pitman Estimators 344 (2)
5.8 Estimating Equations 346 (3)
5.8.1 Moment-Equations Estimators 346 (1)
5.8.2 General Theory of Estimating 347 (2)
Functions
5.9 Pretest Estimators 349 (1)
5.10 Robust Estimation of the Location 349 (57)
and Scale Parameters of Symmetric
Distributions
Part II Examples 353 (28)
Part III Problems 381 (12)
Part IV Solutions Of Selected Problems 393 (13)
6 Confidence and Tolerance Intervals 406 (33)
Part I Theory 406 (1)
6.1 General Introduction 406 (1)
6.2 The Construction of Confidence 407 (1)
Intervals
6.3 Optimal Confidence Intervals 408 (2)
6.4 Tolerance Intervals 410 (2)
6.5 Distribution Free Confidence and 412 (2)
Tolerance Intervals
6.6 Simultaneous Confidence Intervals 414 (3)
6.7 Two-Stage and Sequential Sampling for 417 (22)
Fixed Width Confidence Intervals
Part II Examples 421 (8)
Part III Problems 429 (4)
Part IV Solution To Selected Problems 433 (6)
7 Large Sample Theory for Estimation and 439 (46)
Testing
Part I Theory 439 (1)
7.1 Consistency of Estimators and Tests 439 (1)
7.2 Consistency of the MLE 440 (2)
7.3 Asymptotic Normality and Efficiency 442 (2)
of Consistent Estimators
7.4 Second-Order Efficiency of BAN 444 (1)
Estimators
7.5 Large Sample Confidence Intervals 445 (1)
7.6 Edgeworth and Saddlepoint 446 (2)
Approximations to the Distribution of the
MLE: One-Parameter Canonical Exponential
Families
7.7 Large Sample Tests 448 (1)
7.8 Pitman's Asymptotic Efficiency of 449 (2)
Tests
7.9 Asymptotic Properties of Sample 451 (34)
Quantiles
Part H Examples 454 (21)
Part III Problems 475 (4)
Part IV Solution Of Selected Problems 479 (6)
8 Bayesian Analysis in Testing and 485 (78)
Estimation
Part I Theory 485 (1)
8.1 The Bayesian Framework 486 (5)
8.1.1 Prior, Posterior, and Predictive 486 (1)
Distributions
8.1.2 Noninformative and Improper Prior 487 (2)
Distributions
8.1.3 Risk Functions and Bayes 489 (2)
Procedures
8.2 Bayesian Testing of Hypothesis 491 (10)
8.2.1 Testing Simple Hypothesis 491 (2)
8.2.2 Testing Composite Hypotheses 493 (2)
8.2.3 Bayes Sequential Testing of 495 (6)
Hypotheses
8.3 Bayesian Credibility and Prediction 501 (1)
Intervals
8.3.1 Credibility Intervals 501 (1)
8.3.2 Prediction Intervals 501 (1)
8.4 Bayesian Estimation 502 (4)
8.4.1 General Discussion and Examples 502 (1)
8.4.2 Hierarchical Models 502 (2)
8.4.3 The Normal Dynamic Linear Model 504 (2)
8.5 Approximation Methods 506 (7)
8.5.1 Analytical Approximations 506 (2)
8.5.2 Numerical Approximations 508 (5)
8.6 Empirical Bayes Estimators 513 (50)
Part II Examples 514 (35)
Part III Problems 549 (8)
Part IV Solutions Of Selected Problems 557 (6)
9 Advanced Topics in Estimation Theory 563 (38)
Part I Theory 563 (1)
9.1 Minimax Estimators 563 (2)
9.2 Minimum Risk Equivariant, Bayes 565 (5)
Equivariant, and Structural Estimators
9.2.1 Formal Bayes Estimators for 566 (2)
Invariant Priors
9.2.2 Equivariant Estimators Based on 568 (2)
Structural Distributions
9.3 The Admissibility of Estimators 570 (31)
9.3.1 Some Basic Results 570 (5)
9.3.2 The Inadmissibility of Some 575 (7)
Commonly Used Estimators
9.3.3 Minimax and Admissible Estimators 582 (2)
of the Location Parameter
9.3.4 The Relationship of Empirical 584 (1)
Bayes and Stein-Type Estimators of the
Location Parameter in the Normal Case
Part II Examples 585 (7)
Part III Problems 592 (4)
Part IV Solutions Of Selected Problems 596 (5)
References 601 (12)
Author Index 613 (4)
Subject Index 617