Mathematical Statistics

DIETER RASCH, PhD, is scientific advisor at the Center for Design of Experiments at the University of Natural Resources and Life Sciences, Vienna, Austria. He has published more than 275 scientific papers and 56 books as author or editor. DIETER SCHOTT obtained his PhD in analysis from the University of Rostock in 1976 and did his habilitation in the field of numerical functional analysis in 1982. He has published more than 100 scientific papers and is active as author, co-author and editor of numerous books and scientific journals.

• Preface xiii1 Basic Ideas of Mathematical Statistics 11.1 Statistical Population and Samples 21.1.1 Concrete Samples and Statistical Populations 21.1.2 Sampling Procedures 41.2 Mathematical Models for Population and Sample 81.3 Sufficiency and Completeness 91.4 The Notion of Information in Statistics 201.5 Statistical Decision Theory 281.6 Exercises 32References 372 Point Estimation 392.1 Optimal Unbiased Estimators 412.2 Variance-Invariant Estimation 532.3 Methods for Construction and Improvement of Estimators 572.3.1 Maximum Likelihood Method 572.3.2 Least Squares Method 602.3.3 Minimum Chi-Squared Method 612.3.4 Method of Moments 622.3.5 Jackknife Estimators 632.3.6 Estimators Based on Order Statistics 642.3.6.1 Order and Rank Statistics 642.3.6.2 L-Estimators 662.3.6.3 M-Estimators 672.3.6.4 R-Estimators 682.4 Properties of Estimators 682.4.1 Small Samples 692.4.2 Asymptotic Properties 712.5 Exercises 75References 783 Statistical Tests and Confidence Estimations 793.1 Basic Ideas of Test Theory 793.2 The Neyman–Pearson Lemma 873.3 Tests for Composite Alternative Hypotheses and One-Parametric Distribution Families 963.3.1 Distributions with Monotone Likelihood Ratio and Uniformly Most Powerful Tests for One-Sided Hypotheses 963.3.2 UMPU-Tests for Two-Sided Alternative Hypotheses 1053.4 Tests for Multi-Parametric Distribution Families 1103.4.1 General Theory 1113.4.2 The Two-Sample Problem: Properties of Various Tests and Robustness 1243.4.2.1 Comparison of Two Expectations 1253.4.3 Comparison of Two Variances 1373.4.4 Table for Sample Sizes 1383.5 Confidence Estimation 1393.5.1 One-Sided Confidence Intervals in One-Parametric Distribution Families 1403.5.2 Two-Sided Confidence Intervals in One-Parametric and Confidence Intervals in Multi-Parametric Distribution Families 1433.5.3 Table for Sample Sizes 1463.6 Sequential Tests 1473.6.1 Introduction 1473.6.2 Wald’s Sequential Likelihood Ratio Test for One-Parametric Exponential Families 1493.6.3 Test about Mean Values for Unknown Variances 1533.6.4 Approximate Tests for the Two-Sample Problem 1583.6.5 Sequential Triangular Tests 1603.6.6 A Sequential Triangular Test for the Correlation Coefficient 1623.7 Remarks about Interpretation 1693.8 Exercises 170References 1764 Linear Models – General Theory 1794.1 Linear Models with Fixed Effects 1794.1.1 Least Squares Method 1804.1.2 Maximum Likelihood Method 1844.1.3 Tests of Hypotheses 1854.1.4 Construction of Confidence Regions 1904.1.5 Special Linear Models 1914.1.6 The Generalised Least Squares Method (GLSM) 1984.2 Linear Models with Random Effects: Mixed Models 1994.2.1 Best Linear Unbiased Prediction (BLUP) 2004.2.2 Estimation of Variance Components 2024.3 Exercises 203References 2045 Analysis of Variance (ANOVA) – Fixed Effects Models (Model I of Analysis of Variance) 2075.1 Introduction 2075.2 Analysis of Variance with One Factor (Simple- or One-Way Analysis of Variance) 2155.2.1 The Model and the Analysis 2155.2.2 Planning the Size of an Experiment 2285.2.2.1 General Description for All Sections of This Chapter 2285.2.2.2 The Experimental Size for the One-Way Classification 2315.3 Two-Way Analysis of Variance 2325.3.1 Cross-Classification (A × B) 2335.3.1.1 Parameter Estimation 2365.3.1.2 Testing Hypotheses 2445.3.2 Nested Classification (A B) 2605.4 Three-Way Classification 2725.4.1 Complete Cross-Classification (A × B × C) 2725.4.2 Nested Classification (C ≺B≺A) 2795.4.3 Mixed Classification 2825.4.3.1 Cross-Classification between Two Factors Where One of Them Is Subordinated to a Third Factor B≺a × c 2825.4.3.2 Cross-Classification of Two Factors in Which a Third Factor Is Nested C ≺ A × B 2885.5 Exercises 291References 2916 Analysis of Variance: Estimation of Variance Components (Model II of the Analysis of Variance) 2936.1 Introduction: Linear Models with Random Effects 2936.2 One-Way Classification 2976.2.1 Estimation of Variance Components 3006.2.1.1 Analysis of Variance Method 3006.2.1.2 Estimators in Case of Normally Distributed Y 3026.2.1.3 REML Estimation 3046.2.1.4 Matrix Norm Minimising Quadratic Estimation 3056.2.1.5 Comparison of Several Estimators 3066.2.2 Tests of Hypotheses and Confidence Intervals 3086.2.3 Variances and Properties of the Estimators of the Variance Components 3106.3 Estimators of Variance Components in the Two-Way and Three-Way Classification 3156.3.1 General Description for Equal and Unequal Subclass Numbers 3156.3.2 Two-Way Cross-Classification 3196.3.3 Two-Way Nested Classification 3246.3.4 Three-Way Cross-Classification with Equal Subclass Numbers 3266.3.5 Three-Way Nested Classification 3336.3.6 Three-Way Mixed Classification 3356.4 Planning Experiments 3366.5 Exercises 338References 3397 Analysis of Variance – Models with Finite Level Populations and Mixed Models 3417.1 Introduction: Models with Finite Level Populations 3417.2 Rules for the Derivation of SS, df, MS and E(MS) in Balanced ANOVA Models 3437.3 Variance Component Estimators in Mixed Models 3487.3.1 An Example for the Balanced Case 3497.3.2 The Unbalanced Case 3517.4 Tests for Fixed Effects and Variance Components 3537.5 Variance Component Estimation and Tests of Hypotheses in Special Mixed Models 3547.5.1 Two-Way Cross-Classification 3557.5.2 Two-Way Nested Classification B ≺ A 3587.5.2.1 Levels of A Random 3607.5.2.2 Levels of B Random 3617.5.3 Three-Way Cross-Classification 3627.5.4 Three-Way Nested Classification 3657.5.5 Three-Way Mixed Classification 3687.5.5.1 The Type (B ≺ A)×C 3687.5.5.2 The Type C ≺ AB 3717.6 Exercises 374References 3748 Regression Analysis – Linear Models with Non-random Regressors (Model I of Regression Analysis) and with Random Regressors (Model II of Regression Analysis) 3778.1 Introduction 3778.2 Parameter Estimation 3808.2.1 Least Squares Method 3808.2.2 Optimal Experimental Design 3948.3 Testing Hypotheses 3978.4 Confidence Regions 4068.5 Models with Random Regressors 4108.5.1 Analysis 4108.5.2 Experimental Designs 4158.6 Mixed Models 4168.7 Concluding Remarks about Models of Regression Analysis 4178.8 Exercises 419References 4199 Regression Analysis – Intrinsically Non-linear Model I 4219.1 Estimating by the Least Squares Method 4249.1.1 Gauss–Newton Method 4259.1.2 Internal Regression 4319.1.3 Determining Initial Values for Iteration Methods 4339.2 Geometrical Properties 4349.2.1 Expectation Surface and Tangent Plane 4349.2.2 Curvature Measures 4409.3 Asymptotic Properties and the Bias of LS Estimators 4439.4 Confidence Estimations and Tests 4479.4.1 Introduction 4479.4.2 Tests and Confidence Estimations Based on the Asymptotic Covariance Matrix 4519.4.3 Simulation Experiments to Check Asymptotic Tests and Confidence Estimations 4529.5 Optimal Experimental Design 4549.6 Special Regression Functions 4589.6.1 Exponential Regression 4589.6.1.1 Point Estimator 4589.6.1.2 Confidence Estimations and Tests 4609.6.1.3 Results of Simulation Experiments 4639.6.1.4 Experimental Designs 4669.6.2 The Bertalanffy Function 4689.6.3 The Logistic (Three-Parametric Hyperbolic Tangent) Function 4739.6.4 The Gompertz Function 4769.6.5 The Hyperbolic Tangent Function with Four Parameters 4809.6.6 The Arc Tangent Function with Four Parameters 4849.6.7 The Richards Function 4879.6.8 Summarising the Results of Sections 9.6.1–9.6.7 4879.6.9 Problems of Model Choice 4889.7 Exercises 489References 49010 Analysis of Covariance (ANCOVA) 49510.1 Introduction 49510.2 General Model I–I of the Analysis of Covariance 49610.3 Special Models of the Analysis of Covariance for the Simple Classification 50310.3.1 One Covariable with Constant γ 50410.3.2 A Covariable with Regression Coefficients Γ I Depending on the Levels of the Classification Factor 50610.3.3 A Numerical Example 50710.4 Exercises 510References 51111 Multiple Decision Problems 51311.1 Selection Procedures 51411.1.1 Basic Ideas 51411.1.2 Indifference Zone Formulation for Expectations 51611.1.2.1 Selection of Populations with Normal Distribution 51711.1.2.2 Approximate Solutions for Non-normal Distributions and t =1 52911.1.3 Selection of a Subset Containing the Best Population with Given Probability 53111.1.3.1 Selection of the Normal Distribution with the Largest Expectation 53411.1.3.2 Selection of the Normal Distribution with Smallest Variance 53411.2 Multiple Comparisons 53911.2.1 Confidence Intervals for All Contrasts: Scheffé’s Method 54211.2.2 Confidence Intervals for Given Contrasts: Bonferroni’s and Dunn’s Method 54811.2.3 Confidence Intervals for All Contrasts for N I = N: Tukey’s Method 55011.2.4 Confidence Intervals for All Contrasts: Generalised Tukey’s Method 55311.2.5 Confidence Intervals for the Differences of Treatments with a Control: Dunnett’s Method 55411.2.6 Multiple Comparisons and Confidence Intervals 55611.2.7 Which Multiple Comparison Shall Be Used? 55911.3 A Numerical Example 55911.4 Exercises 563References 56312 Experimental Designs 56712.1 Introduction 56812.2 Block Designs 57112.2.1 Completely Balanced Incomplete Block Designs (BIBD) 57412.2.2 Construction Methods of BIBD 58212.2.3 Partially Balanced Incomplete Block Designs 59612.3 Row–Column Designs 60012.4 Factorial Designs 60312.5 Programs for Construction of Experimental Designs 60412.6 Exercises 604References 605Appendix A: Symbolism 609Appendix B: Abbreviations 611Appendix C: Probability and Density Functions 613Appendix D: Tables 615Solutions and Hints for Exercises 627Index 659