# Statistics of Directional Data

## 1st Edition

**Author:**K. V. Mardia

**Editors:**Z. W. Birnbaum E. Lukacs

**eBook ISBN:**9781483218663

**Imprint:**Academic Press

**Published Date:**28th January 1972

**Page Count:**380

## Description

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Statistics of Directional Data aims to provide a systematic account of statistical theory and methodology for observations which are directions.

The publication first elaborates on angular data and frequency distributions, descriptive measures, and basic concepts and theoretical models. Discussions focus on moments and measures of location and dispersion, distribution function, corrections for grouping, calculation of the mean direction and the circular variance, interrelations between different units of angular measurement, and diagrammatical representation. The book then examines fundamental theorems and distribution theory, point estimation, and tests for samples from von Mises populations.

The text takes a look at non-parametric tests, distributions on spheres, and inference problems on the sphere. Topics include tests for axial data, point estimation, distribution theory, moments and limiting distributions, and tests of goodness of fit and tests of uniformity.

The publication is a dependable reference for researchers interested in probability and mathematical statistics.

## Table of Contents

Preface

Introduction

Chapter 1—Angular Data and Frequency Distributions

1. Introduction

2. Diagrammatical Representation

3. Interrelations Between Different Units of Angular Measurement

4. Forms of Frequency Distributions

5. Further Examples of Angular Data

Chapter 2—Descriptive Measures

1. Introduction

2. A Measure of Location

3. The Circular Variance

4. Calculation of the Mean Direction and the Circular Variance

5. Some Other Measures of Location

6. Some other Measures of Dispersion

7. Trigonometric Moments

8. Corrections for Grouping

Chapter 3—Basic Concepts and Theoretical Models

1. The Distribution Function

2. The Characteristic Function

3. Moments and Measures of Location and Dispersion

4. Circular Models

5. Angular Distributions on the Range (0, 2π/l)

6. Mixtures and Multi-Modal Distributions

7. Circular Standard Deviation, Skewness and Kurtosis

8. Corrections for Grouping

Chapter 4—Fundamental Theorems and Distribution Theory

1. Introduction

2. Theorems on the Characteristic Function

3. Limit Theorems

4. The Isotropic Random Walk on the Circle

5. Distributions of C, S and R for a von Mises Population

6. Distributions Related to the Multi-Sample Problem for von Mises Populations

7. Moments of R

8. The Moments of C and S

9. Limiting Distributions of Angular Statistics

Chapter 5—Point Estimation

1. A Cramér-Rao Type Bound

2. The Method of Moments

3. Sufficiency

4. The von Mises Distribution

5. A Regression Model

6. Mixtures of von Mises Distributions

Chapter 6—Tests for Samples from von Mises Populations

1. Introduction

2. Single Sample Tests

3. Two-Sample Tests

4. Multi-Sample Tests

5. A Regression Model

6. Tests for Multi-Modal and Axial Data

Chapter 7—Non-Parametric Tests

1. Introduction and Basic Results

2. Tests of Goodness of Fit and Tests of Uniformity

3. Tests of Symmetry

4. Two-Sample Tests

5. Multi-Sample Tests

6. Tests for Multimodal and Axial Data

Chapter 8—Distributions on Spheres

1. Spherical Data

2. Other Spherical Co-Ordinate Systems

3. Azimuthal Projections

4. Descriptive Measures

5. Models

6. Distribution Theory

7. Moments and Limiting Distributions

8. A Distribution on a Hypersphere

Chapter 9—Inference Problems on the Sphere

1. Introduction

2. Point Estimation

3. Single Sample Tests

4. Two-Sample Tests

5. Multi-Sample Tests

6. A Test for Coplanarity

7. Tests for Axial Data

8. A Review of Some Other Tests and Topics

Appendix 1—Bessel Functions

Appendix 2—Tables and Charts for the Circular Case (Abridged Titles)

1. The von Mises Distribution Function

2. The Population Resultant Length for the von Mises Case

3. Maximum Likelihood Estimates for The von Mises Case

4. A Test of Uniformity When the Mean Direction is Known

5. Critical Values for the Rayleigh Test (Circular Case)

6. Percentage Points of the von Mises Distribution

7. Confidence Interval for the Mean Direction

8. Confidence Interval for the Concentration Parameter

9. Critical Values for Watson-Williams' Two-Sample Test

10. Critical Values for Kuiper's Test

11. Critical Values for Hodges-Ajne's Test

12. Critical Values of the Circular Range

13. Critical Values for the Equal Spacings Test

14. Critical Values for the Uniform Scores Test

15. Critical Values for Watson's Two-Sample Tests

16. Critical Values for the Run Test

17. Critical Values for the Multi-Sample Uniform Scores Test

18. Critical Values for the Bimodal Scores Test

Appendix 3—Tables and Charts for the Spherical Case (Abridged Titles)

1. Percentage Points of the Fisher Distribution

2. Maximum Likelihood Estimates for the Fisher Case

3. Maximum Likelihood Estimates for the Girdle Case

4. Maximum Likelihood Estimates for the Bipolar Case

5. Critical Values for the Rayleigh Test (Spherical Case)

6. Critical Values for Testing a Prescribed Direction

7. Critical Values for Testing a Prescribed Concentration Parameter

8. Critical Values for Watson-Williams' Two-Sample Test

Bibliography and Author Index

Subject Index

Index of Notation

## Details

- No. of pages:
- 380

- Language:
- English

- Copyright:
- © Academic Press 1972

- Published:
- 28th January 1972

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483218663

## About the Author

### K. V. Mardia

## About the Editors

### Z. W. Birnbaum

### E. Lukacs

### Affiliations and Expertise

Bowling Green State University