# Mathematical Statistics

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# Mathematical Statistics

ISBN: 978-1-119-38523-3 January 2018 688 Pages

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## Description

Explores mathematical statistics in its entirety—from the fundamentals to modern methods

This book introduces readers to point estimation, confidence intervals, and statistical tests. Based on the general theory of linear models, it provides an in-depth overview of the following: analysis of variance (ANOVA) for models with fixed, random, and mixed effects; regression analysis is also first presented for linear models with fixed, random, and mixed effects before being expanded to nonlinear models; statistical multi-decision problems like statistical selection procedures (Bechhofer and Gupta) and sequential tests; and design of experiments from a mathematical-statistical point of view. Most analysis methods have been supplemented by formulae for minimal sample sizes. The chapters also contain exercises with hints for solutions.

Translated from the successful German text, Mathematical Statistics requires knowledge of probability theory (combinatorics, probability distributions, functions and sequences of random variables), which is typically taught in the earlier semesters of scientific and mathematical study courses. It teaches readers all about statistical analysis and covers the design of experiments. The book also describes optimal allocation in the chapters on regression analysis. Additionally, it features a chapter devoted solely to experimental designs.

• Classroom-tested with exercises included
• Practice-oriented (taken from day-to-day statistical work of the authors)
• Includes further studies including design of experiments and sample sizing
• Presents and uses IBM SPSS Statistics 24 for practical calculations of data

Mathematical Statistics is a recommended text for advanced students and practitioners of math, probability, and statistics.

Preface xiii

1 Basic Ideas of Mathematical Statistics 1

1.1 Statistical Population and Samples 2

1.1.1 Concrete Samples and Statistical Populations 2

1.1.2 Sampling Procedures 4

1.2 Mathematical Models for Population and Sample 8

1.3 Sufficiency and Completeness 9

1.4 The Notion of Information in Statistics 20

1.5 Statistical Decision Theory 28

1.6 Exercises 32

References 37

2 Point Estimation 39

2.1 Optimal Unbiased Estimators 41

2.2 Variance-Invariant Estimation 53

2.3 Methods for Construction and Improvement of Estimators 57

2.3.1 Maximum Likelihood Method 57

2.3.2 Least Squares Method 60

2.3.3 Minimum Chi-Squared Method 61

2.3.4 Method of Moments 62

2.3.5 Jackknife Estimators 63

2.3.6 Estimators Based on Order Statistics 64

2.3.6.1 Order and Rank Statistics 64

2.3.6.2 L-Estimators 66

2.3.6.3 M-Estimators 67

2.3.6.4 R-Estimators 68

2.4 Properties of Estimators 68

2.4.1 Small Samples 69

2.4.2 Asymptotic Properties 71

2.5 Exercises 75

References 78

3 Statistical Tests and Confidence Estimations 79

3.1 Basic Ideas of Test Theory 79

3.2 The Neyman–Pearson Lemma 87

3.3 Tests for Composite Alternative Hypotheses and One-Parametric Distribution Families 96

3.3.1 Distributions with Monotone Likelihood Ratio and Uniformly Most Powerful Tests for One-Sided Hypotheses 96

3.3.2 UMPU-Tests for Two-Sided Alternative Hypotheses 105

3.4 Tests for Multi-Parametric Distribution Families 110

3.4.1 General Theory 111

3.4.2 The Two-Sample Problem: Properties of Various Tests and Robustness 124

3.4.2.1 Comparison of Two Expectations 125

3.4.3 Comparison of Two Variances 137

3.4.4 Table for Sample Sizes 138

3.5 Confidence Estimation 139

3.5.1 One-Sided Confidence Intervals in One-Parametric Distribution Families 140

3.5.2 Two-Sided Confidence Intervals in One-Parametric and Confidence Intervals in Multi-Parametric Distribution Families 143

3.5.3 Table for Sample Sizes 146

3.6 Sequential Tests 147

3.6.1 Introduction 147

3.6.2 Wald’s Sequential Likelihood Ratio Test for One-Parametric Exponential Families 149

3.6.3 Test about Mean Values for Unknown Variances 153

3.6.4 Approximate Tests for the Two-Sample Problem 158

3.6.5 Sequential Triangular Tests 160

3.6.6 A Sequential Triangular Test for the Correlation Coefficient 162

3.8 Exercises 170

References 176

4 Linear Models: General Theory 179

4.1 Linear Models with Fixed Effects 179

4.1.1 Least Squares Method 180

4.1.2 Maximum Likelihood Method 184

4.1.3 Tests of Hypotheses 185

4.1.4 Construction of Confidence Regions 190

4.1.5 Special Linear Models 191

4.1.6 The Generalised Least Squares Method (GLSM) 198

4.2 Linear Models with Random Effects: Mixed Models 199

4.2.1 Best Linear Unbiased Prediction (BLUP) 200

4.2.2 Estimation of Variance Components 202

4.3 Exercises 203

References 204

5 Analysis of Variance (ANOVA): Fixed Effects Models (Model I of Analysis of Variance) 207

5.1 Introduction 207

5.2 Analysis of Variance with One Factor (Simple- or One-Way Analysis of Variance) 215

5.2.1 The Model and the Analysis 215

5.2.2 Planning the Size of an Experiment 228

5.2.2.1 General Description for All Sections of This Chapter 228

5.2.2.2 The Experimental Size for the One-Way Classification 231

5.3 Two-Way Analysis of Variance 232

5.3.1 Cross-Classification (A × B) 233

5.3.1.1 Parameter Estimation 236

5.3.1.2 Testing Hypotheses 244

5.3.2 Nested Classification (A B) 260

5.4 Three-Way Classification 272

5.4.1 Complete Cross-Classification (A × B × C) 272

5.4.2 Nested Classification (C≺B≺A) 279

5.4.3 Mixed Classification 282

5.4.3.1 Cross-Classification between Two Factors Where One of Them Is Subordinated to a Third Factor B≺A × C 282

5.4.3.2 Cross-Classification of Two Factors in Which a Third Factor Is Nested C≺ A× B 288

5.5 Exercises 291

References 291

6 Analysis of Variance: Estimation of Variance Components (Model II of the Analysis of Variance) 293

6.1 Introduction: Linear Models with Random Effects 293

6.2 One-Way Classification 297

6.2.1 Estimation of Variance Components 300

6.2.1.1 Analysis of Variance Method 300

6.2.1.2 Estimators in Case of Normally Distributed Y 302

6.2.1.3 REML: Estimation 304

6.2.1.4 Matrix Norm Minimising Quadratic Estimation 305

6.2.1.5 Comparison of Several Estimators 306

6.2.2 Tests of Hypotheses and Confidence Intervals 308

6.2.3 Variances and Properties of the Estimators of the Variance Components 310

6.3 Estimators of Variance Components in the Two-Way and Three-Way Classification 315

6.3.1 General Description for Equal and Unequal Subclass Numbers 315

6.3.2 Two-Way Cross-Classification 319

6.3.3 Two-Way Nested Classification 324

6.3.4 Three-Way Cross-Classification with Equal Subclass Numbers 326

6.3.5 Three-Way Nested Classification 334

6.3.6 Three-Way Mixed Classification 334

6.4 Planning Experiments 336

6.5 Exercises 338

References 339

7 Analysis of Variance: Models with Finite Level Populations and Mixed Models 341

7.1 Introduction: Models with Finite Level Populations 341

7.2 Rules for the Derivation of SS, df, MS and E(MS) in Balanced ANOVA Models 343

7.3 Variance Component Estimators in Mixed Models 348

7.3.1 An Example for the Balanced Case 349

7.3.2 The Unbalanced Case 351

7.4 Tests for Fixed Effects and Variance Components 353

7.5 Variance Component Estimation and Tests of Hypotheses in Special Mixed Models 354

7.5.1 Two-Way Cross-Classification 355

7.5.2 Two-Way Nested Classification B ≺ A 358

7.5.2.1 Levels of A Random 360

7.5.2.2 Levels of B Random 361

7.5.3 Three-Way Cross-Classification 362

7.5.4 Three-Way Nested Classification 365

7.5.5 Three-Way Mixed Classification 369

7.5.5.1 The Type (B ≺ A) × C 369

7.5.5.2 The Type C ≺ AB 374

7.6 Exercises 376

References 376

8 Regression Analysis: Linear Models with Non-random Regressors (Model I of Regression Analysis) and with Random Regressors (Model II of Regression Analysis) 377

8.1 Introduction 377

8.2 Parameter Estimation 380

8.2.1 Least Squares Method 380

8.2.2 Optimal Experimental Design 394

8.3 Testing Hypotheses 397

8.4 Confidence Regions 406

8.5 Models with Random Regressors 410

8.5.1 Analysis 410

8.5.2 Experimental Designs 415

8.6 Mixed Models 416

8.7 Concluding Remarks about Models of Regression Analysis 417

8.8 Exercises 419

References 419

9 Regression Analysis: Intrinsically Non-linear Model I 421

9.1 Estimating by the Least Squares Method 424

9.1.1 Gauß–Newton Method 425

9.1.2 Internal Regression 431

9.1.3 Determining Initial Values for Iteration Methods 433

9.2 Geometrical Properties 434

9.2.1 Expectation Surface and Tangent Plane 434

9.2.2 Curvature Measures 440

9.3 Asymptotic Properties and the Bias of LS Estimators 443

9.4 Confidence Estimations and Tests 447

9.4.1 Introduction 447

9.4.2 Tests and Confidence Estimations Based on the Asymptotic Covariance Matrix 451

9.4.3 Simulation Experiments to Check Asymptotic Tests and Confidence Estimations 452

9.5 Optimal Experimental Design 454

9.6 Special Regression Functions 458

9.6.1 Exponential Regression 458

9.6.1.1 Point Estimator 458

9.6.1.2 Confidence Estimations and Tests 460

9.6.1.3 Results of Simulation Experiments 463

9.6.1.4 Experimental Designs 466

9.6.2 The Bertalanffy Function 468

9.6.3 The Logistic (Three-Parametric Hyperbolic Tangent) Function 473

9.6.4 The Gompertz Function 476

9.6.5 The Hyperbolic Tangent Function with Four Parameters 479

9.6.6 The arc tangent Function with Four Parameters 484

9.6.7 The Richards Function 487

9.6.8 Summarising the Results of Sections 9.6.1–9.6.7 487

9.6.9 Problems of Model Choice 488

9.7 Exercises 489

References 490

10 Analysis of Covariance (ANCOVA) 495

10.1 Introduction 495

10.2 General Model I–I of the Analysis of Covariance 496

10.3 Special Models of the Analysis of Covariance for the Simple Classification 503

10.3.1 One Covariable with Constant γ 504

10.3.2 A Covariable with Regression Coefficients γi Depending on the Levels of the Classification Factor 506

10.3.3 A Numerical Example 507

10.4 Exercises 510

References 511

11 Multiple Decision Problems 513

11.1 Selection Procedures 514

11.1.1 Basic Ideas 514

11.1.2 Indifference Zone Formulation for Expectations 516

11.1.2.1 Selection of Populations with Normal Distribution 517

11.1.2.2 Approximate Solutions for Non-normal Distributions and t = 1 529

11.1.3 Selection of a Subset Containing the Best Population with Given Probability 530

11.1.3.1 Selection of the Normal Distribution with the Largest Expectation 534

11.1.3.2 Selection of the Normal Distribution with Smallest Variance 535

11.2 Multiple Comparisons 536

11.2.1 Confidence Intervals for All Contrasts: Scheffé’s Method 542

11.2.2 Confidence Intervals for Given Contrast: Bonferroni’s and Dunn’s Method 547

11.2.3 Confidence Intervals for All Contrasts for ni = n: Tukey’s Method 550

11.2.4 Confidence Intervals for All Contrast: Generalised Tukey’s Method 553

11.2.5 Confidence Intervals for the Differences of Treatments with a Control: Dunnett’s Method 555

11.2.6 Multiple Comparisons and Confidence Intervals 556

11.2.7 Which Multiple Comparisons Shall Be Used? 559

11.3 A Numerical Example 560

11.4 Exercises 564

References 564

12 Experimental Designs 567

12.1 Introduction 568

12.2 Block Designs 571

12.2.1 Completely Balanced Incomplete Block Designs (BIBD) 574

12.2.2 Construction Methods of BIBD 582

12.2.3 Partially Balanced Incomplete Block Designs 596

12.3 Row–Column Designs 600

12.4 Factorial Designs 603

12.5 Programs for Construction of Experimental Designs 604

12.6 Exercises 604

References 605

Appendix A: Symbolism 609

Appendix B: Abbreviations 611

Appendix C: Probability and Density Functions 613

Appendix D: Tables 615

Solutions and Hints for Exercises 627

Index 659