O'Reilly logo

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Mathematical Statistics

Book 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.

 

Table of Contents

  1. Cover
  2. Title Page
  3. Preface
    1. References
  4. 1 Basic Ideas of Mathematical Statistics
    1. 1.1 Statistical Population and Samples
    2. 1.2 Mathematical Models for Population and Sample
    3. 1.3 Sufficiency and Completeness
    4. 1.4 The Notion of Information in Statistics
    5. 1.5 Statistical Decision Theory
    6. 1.6 Exercises
    7. References
  5. 2 Point Estimation
    1. 2.1 Optimal Unbiased Estimators
    2. 2.2 Variance‐Invariant Estimation
    3. 2.3 Methods for Construction and Improvement of Estimators
    4. 2.4 Properties of Estimators
    5. 2.5 Exercises
    6. References
  6. 3 Statistical Tests and Confidence Estimations
    1. 3.1 Basic Ideas of Test Theory
    2. 3.2 The Neyman–Pearson Lemma
    3. 3.3 Tests for Composite Alternative Hypotheses and One‐Parametric Distribution Families
    4. 3.4 Tests for Multi‐Parametric Distribution Families
    5. 3.5 Confidence Estimation
    6. 3.6 Sequential Tests
    7. 3.7 Remarks about Interpretation
    8. 3.8 Exercises
    9. References
  7. 4 Linear Models – General Theory
    1. 4.1 Linear Models with Fixed Effects
    2. 4.2 Linear Models with Random Effects: Mixed Models
    3. 4.3 Exercises
    4. References
  8. 5 Analysis of Variance (ANOVA) – Fixed Effects Models (Model I of Analysis of Variance)
    1. 5.1 Introduction
    2. 5.2 Analysis of Variance with One Factor (Simple‐ or One‐Way Analysis of Variance)
    3. 5.3 Two‐Way Analysis of Variance
    4. 5.4 Three‐Way Classification
    5. 5.5 Exercises
    6. References
  9. 6 Analysis of Variance: Estimation of Variance Components (Model II of the Analysis of Variance)
    1. 6.1 Introduction: Linear Models with Random Effects
    2. 6.2 One‐Way Classification
    3. 6.3 Estimators of Variance Components in the Two‐Way and Three‐Way Classification
    4. 6.4 Planning Experiments
    5. 6.5 Exercises
    6. References
  10. 7 Analysis of Variance – Models with Finite Level Populations and Mixed Models
    1. 7.1 Introduction: Models with Finite Level Populations
    2. 7.2 Rules for the Derivation of SS, df, MS and E(MS) in Balanced ANOVA Models
    3. 7.3 Variance Component Estimators in Mixed Models
    4. 7.4 Tests for Fixed Effects and Variance Components
    5. 7.5 Variance Component Estimation and Tests of Hypotheses in Special Mixed Models
    6. 7.6 Exercises
    7. References
  11. 8 Regression Analysis – Linear Models with Non‐random Regressors (Model I of Regression Analysis) and with Random Regressors (Model II of Regression Analysis)
    1. 8.1 Introduction
    2. 8.2 Parameter Estimation
    3. 8.3 Testing Hypotheses
    4. 8.4 Confidence Regions
    5. 8.5 Models with Random Regressors
    6. 8.6 Mixed Models
    7. 8.7 Concluding Remarks about Models of Regression Analysis
    8. 8.8 Exercises
    9. References
  12. 9 Regression Analysis – Intrinsically Non‐linear Model I
    1. 9.1 Estimating by the Least Squares Method
    2. 9.2 Geometrical Properties
    3. 9.3 Asymptotic Properties and the Bias of LS Estimators
    4. 9.4 Confidence Estimations and Tests
    5. 9.5 Optimal Experimental Design
    6. 9.6 Special Regression Functions
    7. 9.7 Exercises
    8. References
  13. 10 Analysis of Covariance (ANCOVA)
    1. 10.1 Introduction
    2. 10.2 General Model I–I of the Analysis of Covariance
    3. 10.3 Special Models of the Analysis of Covariance for the Simple Classification
    4. 10.4 Exercises
    5. References
  14. 11 Multiple Decision Problems
    1. 11.1 Selection Procedures
    2. 11.2 Multiple Comparisons
    3. 11.3 A Numerical Example
    4. 11.4 Exercises
    5. References
  15. 12 Experimental Designs
    1. 12.1 Introduction
    2. 12.2 Block Designs
    3. 12.3 Row–Column Designs
    4. 12.4 Factorial Designs
    5. 12.5 Programs for Construction of Experimental Designs
    6. 12.6 Exercises
    7. References
  16. Appendix A: Symbolism
  17. Appendix B: Abbreviations
  18. Appendix C: Probability and Density Functions
  19. Appendix D: Tables
  20. Solutions and Hints for Exercises
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  21. Index Mathematical Statistics
  22. End User License Agreement