Mathematical Statistics - Dieter Rasch

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Résumé :

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...

Biographie:

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....

Sommaire:

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

3.7 Remarks about Interpretation 169

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 Analys...