Rasch D., Schott D. Mathematical Statistics

Wiley, 2018. — 688 p. — ISBN: 1119385288.Explores mathematical statistics in its entirety”from the fundamentals to modern methodsThis 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

Basic Ideas of Mathematical Statistics
Statistical Population and Samples
Concrete Samples and Statistical Populations
Sampling Procedures
Mathematical Models for Population and Sample
Sufficiency and Completeness
The Notion of Information in Statistics
Statistical Decision Theory
ExercisesPoint Estimation
Optimal Unbiased Estimators
Variance-Invariant Estimation
Methods for Construction and Improvement of Estimators
Maximum Likelihood Method
Least Squares Method
Minimum Chi-Squared Method
Method of Moments
Jackknife Estimators
Estimators Based on Order Statistics
Order and Rank Statistics
L-Estimators
M-Estimators
R-Estimators
Properties of Estimators
Small Samples
Asymptotic Properties
ExercisesStatistical Tests and Confidence Estimations
Basic Ideas of Test Theory
The Neyman – Pearson Lemma
Tests for Composite Alternative Hypotheses and One-Parametric
Distribution Families
Distributions with Monotone Likelihood Ratio and Uniformly Most
Powerful Tests for One-Sided Hypotheses
UMPU-Tests for Two-Sided Alternative Hypotheses
Tests for Multi-Parametric Distribution Families
General Theory
The Two-Sample Problem: Properties of Various Tests and
Robustness
Comparison of Two Expectations
Comparison of Two Variances
Table for Sample Sizes
Confidence Estimation
One-Sided Confidence Intervals in One-Parametric Distribution Families
Two-Sided Confidence Intervals in One-Parametric and Confidence
Intervals in Multi-Parametric Distribution Families
Table for Sample Sizes
Sequential TestsWald’s Sequential Likelihood Ratio Test for One-Parametric
Exponential Families
Test about Mean Values for Unknown Variances
Approximate Tests for the Two-Sample Problem
Sequential Triangular Tests
A Sequential Triangular Test for the Correlation Coefficient
Remarks about Interpretation
ExercisesLinear Models – General Theory
Linear Models with Fixed Effects
Least Squares Method
Maximum Likelihood Method
Tests of Hypotheses
Construction of Confidence Regions
Special Linear Models
The Generalized Least Squares Method (GLSM)
Linear Models with Random Effects: Mixed Models
Best Linear Unbiased Prediction (BLUP)
Estimation of Variance Components
ExercisesAnalysis of Variance (ANOVA) – Fixed Effects Models (Model I of Analysis of Variance)Analysis of Variance with One Factor (Simple- or One-Way Analysis of Variance)
The Model and the Analysis
Planning the Size of an Experiment
General Description for All Sections of This Chapter
The Experimental Size for the One-Way Classification
Two-Way Analysis of Variance
Cross-Classification (A x B)
Parameter Estimation
Testing Hypotheses
Nested Classification (A> B)
Three-Way Classification
Complete Cross-Classification (A < B < C)
Nested Classification (C<B<A)
Mixed Classification
Cross-Classification between Two Factors Where One of Them Subordinated to a Third Factor ((B<A)x C)
Cross-Classification of Two Factors in Which a Third Factor Is
Nested C A B
ExercisesAnalysis of Variance: Estimation of Variance Components
(Model II of the Analysis of Variance)
Introduction: Linear Models with Random Effects
One-Way Classification
Estimation of Variance Components
Analysis of Variance Method
Estimators in Case of Normally Distributed Y
REML Estimation
Matrix Norm Minimizing Quadratic Estimation
Comparison of Several Estimators
Tests of Hypotheses and Confidence Intervals
Variances and Properties of the Estimators of the Variance
Components
Estimators of Variance Components in the Two-Way and
Three-Way Classification
General Description for Equal and Unequal Subclass Numbers
Two-Way Cross-Classification
Two-Way Nested Classification
Three-Way Cross-Classification with Equal
Subclass Numbers
Three-Way Nested Classification
Three-Way Mixed Classification
Planning Experiments
ExercisesAnalysis of Variance – Models with Finite Level Populations and Mixed Models
Introduction: Models with Finite Level Populations
Rules for the Derivation of SS, df, MS and E(MS) in Balanced
ANOVA Models
Variance Component Estimators in Mixed Models
An Example for the Balanced Case
The Unbalanced Case
Tests for Fixed Effects and Variance Components
Variance Component Estimation and Tests of Hypotheses in
Special Mixed Models
Two-Way Cross-Classification
Two-Way Nested Classification B? A
Levels of A Random
Levels of B Random
Three-Way Cross-Classification
Three-Way Nested Classification
Three-Way Mixed Classification
The Type (B < A) x C
The Type C< AB
ExercisesRegression Analysis – Linear Models with Non-random Regressors (Model I of Regression Analysis) and with Random Regressors (Model II of Regression Analysis)Parameter Estimation
Least Squares Method
Optimal Experimental Design
Testing Hypotheses
Confidence Regions
Models with Random Regressors
Analysis
Experimental Designs
Mixed Models
Concluding Remarks about Models of Regression Analysis
ExercisesRegression Analysis – Intrinsically Non-linear Model I
Estimating by the Least Squares Method
Gauss – Newton Method
Internal Regression
Determining Initial Values for Iteration Methods
Geometrical Properties
Expectation Surface and Tangent Plane
Curvature Measures
Asymptotic Properties and the Bias of LS Estimators
Confidence Estimations and TestsTests and Confidence Estimations Based on the Asymptotic
Covariance Matrix
Simulation Experiments to Check Asymptotic Tests and
Confidence Estimations
Optimal Experimental Design
Special Regression Functions
Exponential Regression
Point Estimator
Confidence Estimations and Tests
Results of Simulation Experiments
Experimental Designs
The Bertalanffy Function
The Logistic (Three-Parametric Hyperbolic Tangent) Function
The Gompertz Function
The Hyperbolic Tangent Function with Four Parameters
The Arc Tangent Function with Four Parameters
The Richards Function
Summarizing the Results of Sections.
Problems of Model Choice
ExercisesAnalysis of Covariance (ANCOVA)General Model I – I of the Analysis of Covariance
Special Models of the Analysis of Covariance for the Simple
Classification
One Covariable with Constant?
A Covariable with Regression Coefficients?i Depending on the Levels
of the Classification Factor
A Numerical Example
ExercisesMultiple Decision Problems
Selection Procedures
Basic Ideas
Indifference Zone Formulation for Expectations
Selection of Populations with Normal Distribution
Approximate Solutions for Non-normal Distributions and t =
Selection of a Subset Containing the Best Population with Given
Probability
Selection of the Normal Distribution with the Largest
Expectation
Selection of the Normal Distribution with Smallest Variance
Multiple Comparisons
Confidence Intervals for All Contrasts: Scheffe’s Method
Confidence Intervals for Given Contrasts: Bonferroni’s and Dunn’s Method
Confidence Intervals for All Contrasts for ni = n: Tukey’s Method
Confidence Intervals for All Contrasts: Generalized Tukey’s Method
Confidence Intervals for the Differences of Treatments with a Control:
Dunnett’s Method
Multiple Comparisons and Confidence Intervals
Which Multiple Comparison Shall Be Used?
A Numerical Example
ExercisesExperimental DesignsBlock Designs
Completely Balanced Incomplete Block Designs (BIBD)
Construction Methods of BIBD
Partially Balanced Incomplete Block Designs
Row – Column Designs
Factorial Designs
Programs for Construction of Experimental Designs
ExercisesAppendix A: Symbolism
Appendix B: Abbreviations
Appendix C: Probability and Density Functions
Appendix D: Tables
Solutions and Hints for Exercises
Index