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**Directional statistics.
2nd ed.**
*(English)*
Zbl 0935.62065

Wiley Series in Probability and Statistics. Chichester: Wiley. xxi, 429 p. (2000).

[For the review of the first edition from 1972 by K. V. Mardia see Zbl 0244.62005.]

The aim of this book is to give a comprehensive, systematic and unified treatment of the theory and methodology of directional statistics. The theory underlying each technique is given and applications by working through real-life examples are illustrated. Three basic approaches (embedding, wrapping and intrinsic approaches) to directional statistics are highlighted throughout. The book has 14 chapters which are classed into three parts. The first part (Chapters 1-8) is concerned with statistics on the circle. Following a general discussion of circular data in Chapter 1, various summary statistics are introduced in Chapter 2.

Chapter 3 gives some basic theoretical concepts and various models for circular data. Distribution functions and characteristic function of distributions on the circle are introduced. Trigonometric moments and various measures of location and dispersion are considered. The key model for circular data consists of the von Mises distributions, which are described fully. The von Mises distributions can be obtained by conditioning bivariate normal distributions. Various other distributions on the circle arise by radial projection of distributions in the plane, or by wrapping distributions from the line to the circle. Some of these distributions are considered.

Chapter 4 gives the fundamental theorems and distribution theory for the uniform and von Mises distributions. Key properties of their characteristic functions are presented. This chapter shows how the usual characteristic function of a random vector in the plane can be used to obtain the distributions of the polar coordinates of this vector. This provides a method of calculating the distributions of the sample mean direction and resultant length of a sample from a distribution on the circle. Some limit theorems on the circle are also considered.

Chapter 5 treats point estimation, mainly for the von Mises and wrapped Cauchy distributions. Because of the special nature of the circle, the concept of unbiased estimation of circular parameters requires careful definition. Such a definition and an appropriate analogue of the CramĂ©r-Rao bound are given. Point estimation for von Mises and wrapped Cauchy distributions is discussed. Interval estimation and robust estimation on the circle are considered in the more general context.

Chapter 6 devotes to the uniform distribution. Because of the central role played by the uniform distribution, one of the most important hypotheses about a distribution on the circle is that of uniformity. Graphical assessment of uniformity and the main formal tests of uniformity are presented. In this chapter, the authors also show how tests of uniformity give rise to tests of goodness-of-fit. Chapter 7 discusses inference on von Mises distributions. Tests and confidence intervals based on a single sample, analogous to those of standard normal theory, are discussed. Two-sample inference and some extensions to the multi-sample case are considered. A test of von Misesness is also described.

Chapter 8 treats nonparametric methods. Nonparametric methods on the circle are based on cumulative distribution functions and ranks. Tests of symmetry and two-sample tests are considered. In particular, two-sample versions of Kuiper’s test and Watson’s \(U\) test are discussed. Extensions to the multi-sample case are given.

The second part (Chapters 9-12) considers statistics on spheres of arbitrary dimension. Chapter 9 discusses several distributions on spheres. Chapter 10 considers estimation for the von Mises-Fisher, Watson, Bingham, Kent and angular central Gaussian distributions. Hypothesis testing for von Mises-Fisher distributions, although extensions to other distributions with rotational symmetry, are considered. Single-sample, two-sample and multi-sample tests are discussed. Chapter 11 is concerned with relationships between directional random variables. Many correlation coefficients have been proposed for measuring the strengths of such relationships. The most important correlation coefficients are presented. Various regression models for describing these relationships are also given. Chapter 12 describes some modern methodology.

The third part (Chapters 13-14) considers extensions to statistics on more general sample spaces. The book contains 4 appendices which give some useful formulae and tables for use in the analysis of circular and spherical data.

The book succeeds as a very good theoretical reference in the study of analysis of circular and spherical data and will be useful for researchers in probability and statistics.

The aim of this book is to give a comprehensive, systematic and unified treatment of the theory and methodology of directional statistics. The theory underlying each technique is given and applications by working through real-life examples are illustrated. Three basic approaches (embedding, wrapping and intrinsic approaches) to directional statistics are highlighted throughout. The book has 14 chapters which are classed into three parts. The first part (Chapters 1-8) is concerned with statistics on the circle. Following a general discussion of circular data in Chapter 1, various summary statistics are introduced in Chapter 2.

Chapter 3 gives some basic theoretical concepts and various models for circular data. Distribution functions and characteristic function of distributions on the circle are introduced. Trigonometric moments and various measures of location and dispersion are considered. The key model for circular data consists of the von Mises distributions, which are described fully. The von Mises distributions can be obtained by conditioning bivariate normal distributions. Various other distributions on the circle arise by radial projection of distributions in the plane, or by wrapping distributions from the line to the circle. Some of these distributions are considered.

Chapter 4 gives the fundamental theorems and distribution theory for the uniform and von Mises distributions. Key properties of their characteristic functions are presented. This chapter shows how the usual characteristic function of a random vector in the plane can be used to obtain the distributions of the polar coordinates of this vector. This provides a method of calculating the distributions of the sample mean direction and resultant length of a sample from a distribution on the circle. Some limit theorems on the circle are also considered.

Chapter 5 treats point estimation, mainly for the von Mises and wrapped Cauchy distributions. Because of the special nature of the circle, the concept of unbiased estimation of circular parameters requires careful definition. Such a definition and an appropriate analogue of the CramĂ©r-Rao bound are given. Point estimation for von Mises and wrapped Cauchy distributions is discussed. Interval estimation and robust estimation on the circle are considered in the more general context.

Chapter 6 devotes to the uniform distribution. Because of the central role played by the uniform distribution, one of the most important hypotheses about a distribution on the circle is that of uniformity. Graphical assessment of uniformity and the main formal tests of uniformity are presented. In this chapter, the authors also show how tests of uniformity give rise to tests of goodness-of-fit. Chapter 7 discusses inference on von Mises distributions. Tests and confidence intervals based on a single sample, analogous to those of standard normal theory, are discussed. Two-sample inference and some extensions to the multi-sample case are considered. A test of von Misesness is also described.

Chapter 8 treats nonparametric methods. Nonparametric methods on the circle are based on cumulative distribution functions and ranks. Tests of symmetry and two-sample tests are considered. In particular, two-sample versions of Kuiper’s test and Watson’s \(U\) test are discussed. Extensions to the multi-sample case are given.

The second part (Chapters 9-12) considers statistics on spheres of arbitrary dimension. Chapter 9 discusses several distributions on spheres. Chapter 10 considers estimation for the von Mises-Fisher, Watson, Bingham, Kent and angular central Gaussian distributions. Hypothesis testing for von Mises-Fisher distributions, although extensions to other distributions with rotational symmetry, are considered. Single-sample, two-sample and multi-sample tests are discussed. Chapter 11 is concerned with relationships between directional random variables. Many correlation coefficients have been proposed for measuring the strengths of such relationships. The most important correlation coefficients are presented. Various regression models for describing these relationships are also given. Chapter 12 describes some modern methodology.

The third part (Chapters 13-14) considers extensions to statistics on more general sample spaces. The book contains 4 appendices which give some useful formulae and tables for use in the analysis of circular and spherical data.

The book succeeds as a very good theoretical reference in the study of analysis of circular and spherical data and will be useful for researchers in probability and statistics.

Reviewer: Wang Songgui (Beijing)

### MSC:

62H11 | Directional data; spatial statistics |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62H10 | Multivariate distribution of statistics |