##
**Statistics on special manifolds.**
*(English)*
Zbl 1026.62051

Lecture Notes in Statistics. 174. New York, NY: Springer. xxvi, 399 p. (2003).

The special manifolds of interest in this book are the Stiefel manifold and the Grassmann manifold. Formally, the Stiefel manifold \(V_{k,m}\) is the space of \(k\)-frames in the \(m\)-dimensional real Euclidean space \(\mathbb R^m\), represented by the set of \(m\times k\) matrices \(X\) such that \(X'X = I_k\), where \(I_k\) is the \(k\times k\) identity matrix, and the Grassmann manifold \(G_{k,m-k}\) is the space of \(k\)-planes (\(k\)-dimensional hyperplanes) in \(\mathbb R^m\). The manifold \(P_{k,m-k}\) of \(m\times m\) orthogonal projection matrices idempotent of rank \(k\) corresponds uniquely to \(G_{k,m-k}\). This book is concerned with statistical analysis on the manifolds \(V_{k,m}\) and \(P_{k,m-k}\) as statistical sample spaces consisting of matrices. The discussion is carried out on the real spaces so that scalars, vectors, and matrices treated in this book are all real, unless explicitly stated otherwise. For the special case \(k= 1\), the observations from \(V_{l,m}\) and \(G_{1.m-1}\) are regarded as directed vectors on a unit sphere and as undirected axes or lines, respectively.

There exists a large literature of applications of directional statistics and its statistical analysis, mostly occurring for \(m = 2\) or 3 in practice, in the Earth (or Geological) Sciences, Astrophysics, Medicine, Biology, Meteorology, Animal Behavior, and many other fields. Examples of observations on the general Grassmann manifold \(G_{k,m-k}\) arise in signal processing of radars with \(m\) elements observing \(k\) targets. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on Grassmann manifolds, which is one of the purposes of this book, must make some contributions to the study in the related sciences.

Ch. 1, The Special Manifolds and Related Multivariate Topics, presents fundamental material which may be helpful for reading the main part of the book: the backgrounds of the special manifolds, examples of orientation statistics in practical problems, and some multivariate calculation techniques and matrix-variate distributions. Ch. 2, Distributions on the Special Manifolds, discusses population distributions, uniform and non-uniform, on the special manifolds. Among those distributions, the matrix Langevin distributions defined on the two manifolds are used for most of the statistical analyses treated in later chapters. This chapter also looks at methods to generate some families of non-uniform distributions, that is, distributions of the orientation and the orthogonal projection matrix of a random rectangular matrix, and further suggests some simulation methods for generating pseudo-random matrices on the manifolds.

Ch. 3, Decompositions of the Special Manifolds, derives various types of decompositions (or transformations) of random matrices and the corresponding decompositions of the invariant measures constructed in Ch. 1 (or Jacobians of the transformations). The results are not only of theoretical interest in themselves, but they are also of practical use for solving some distributional and inferential problems. In Ch. 4, Distributional Problems in the Decomposition Theorems and the Sampling Theory, the decompositions obtained in Ch. 3 are used to derive various distributional results and to introduce general families of distributions on the special manifolds. Various sampling distributions are derived for the sample matrix sums, which are sufficient statistics, taken from the matrix Langevin distributions on the two manifolds. The forms of the sampling distributions are expressed in integral forms involving hypergeometric functions with matrix arguments, which seem to be intractable in themselves. In Ch. 5, The Inference on the Parameters of the Matrix Langevin Distributions, problems of estimation and tests for hypotheses on the parameters are dealt with by Fisher profile scoring methods. These solutions are given in terms of hypergeometric functions with matrix arguments. Bayes estimators, further optimality properties of orientation parameter estimation, and the problem of sufficiency and ancillarity are discussed. In Ch. 6, Large Sample Asymptotic Theorems in Connection with Tests for Uniformity, asymptotic expansions for the density functions of the standardized sample mean matrices and related statistics are derived, and asymptotic properties, near the uniformity, of parameter estimation, and of some optimal tests for uniformity and the asymptotic equivalence of these tests are discussed. Ch. 7, Asymptotic Theorems for Concentrated Matrix Langevin Distributions, is concerned with asymptotic theory for the problem of estimation, sampling distributions, and classification for the concentrated matrix Langevin distributions. The methods are developed to approximately evaluate the estimators of large concentration parameters for each distribution.

Ch. 8, High Dimensional Asymptotic Theorems, investigates the high dimensional asymptotic behavior of matrix statistics and related functions constructed from some main population distributions defined on the special manifolds. The author derives asymptotic expansions for the distributions, generalizes the Stam limit theorems, and discusses asymptotic properties of parameter estimation and tests of hypotheses for the matrix Langevin distributions. Ch. 9, Procrustes Analysis on the Special Manifolds, presents theoretical results obtained by applying Procrustes methods to some statistical analyses on the special manifolds. Procrustes representations of the manifolds and related Procrustes statistics and means are discussed by ordinary, weighted, and generalized Procrustes methods. A brief discussion is given of the isometric and equivariant embeddings of the Stiefel and Grassmann manifolds as spaces defined by the Procrustes representations in Euclidean spaces.

Ch. 10, Density Estimation on the Special Manifolds, develops the theory of density estimation on the special manifolds, where some of the decomposition theorems derived in Ch. 3 play useful roles for the derivations. The theory of density estimation on real spaces of symmetric matrices and of rectangular matrices is developed. Ch. 11, Measures of Orthogonal Association on the Special Manifolds, is concerned with the measurement of orthogonal association on the manifolds, corresponding to linear dependence on the Euclidean space. Population measures of orthogonal association are defined. In particular, on the Stiefel manifold, the author examines the measures of orthogonal association for a family of associated joint distributions, and investigates the asymptotic behavior of the sample measure of orthogonal association for concentrated matrix Langevin distributions. The related problem of orientational regressions is also discussed.

In Appendix A, Invariant Polynomials with Matrix Arguments, a survey of the theory of zonal polynomials (with one matrix argument), including hypergeometric functions with matrix arguments and invariant polynomials with multiple matrix arguments, is presented in consideration of the applications in the text. Appendix B, Generalized Hermite and Laguerre Polynomials with Matrix Arguments, introduces generalized Hermite polynomials and generalized (noncentral) Laguerre polynomials with multiple matrix arguments and discusses various properties of these polynomials, which are useful for the derivations throughout the book. In Appendix C, Edgeworth and Saddle-Point Expansions for Random Matrices, the author develops the methods for obtaining asymptotic expansions for density functions of sample means for random matrices, which are of the Edgeworth, saddle-point, and generalized Edgeworth types, extending the methods for scalar and vector variates.

This book is designed as a reference book for both theoretical and applied statisticians. The book can also be used as a textbook for a graduate course in multivariate analysis for students who specialize in mathematical statistics or multivariate analysis.

There exists a large literature of applications of directional statistics and its statistical analysis, mostly occurring for \(m = 2\) or 3 in practice, in the Earth (or Geological) Sciences, Astrophysics, Medicine, Biology, Meteorology, Animal Behavior, and many other fields. Examples of observations on the general Grassmann manifold \(G_{k,m-k}\) arise in signal processing of radars with \(m\) elements observing \(k\) targets. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on Grassmann manifolds, which is one of the purposes of this book, must make some contributions to the study in the related sciences.

Ch. 1, The Special Manifolds and Related Multivariate Topics, presents fundamental material which may be helpful for reading the main part of the book: the backgrounds of the special manifolds, examples of orientation statistics in practical problems, and some multivariate calculation techniques and matrix-variate distributions. Ch. 2, Distributions on the Special Manifolds, discusses population distributions, uniform and non-uniform, on the special manifolds. Among those distributions, the matrix Langevin distributions defined on the two manifolds are used for most of the statistical analyses treated in later chapters. This chapter also looks at methods to generate some families of non-uniform distributions, that is, distributions of the orientation and the orthogonal projection matrix of a random rectangular matrix, and further suggests some simulation methods for generating pseudo-random matrices on the manifolds.

Ch. 3, Decompositions of the Special Manifolds, derives various types of decompositions (or transformations) of random matrices and the corresponding decompositions of the invariant measures constructed in Ch. 1 (or Jacobians of the transformations). The results are not only of theoretical interest in themselves, but they are also of practical use for solving some distributional and inferential problems. In Ch. 4, Distributional Problems in the Decomposition Theorems and the Sampling Theory, the decompositions obtained in Ch. 3 are used to derive various distributional results and to introduce general families of distributions on the special manifolds. Various sampling distributions are derived for the sample matrix sums, which are sufficient statistics, taken from the matrix Langevin distributions on the two manifolds. The forms of the sampling distributions are expressed in integral forms involving hypergeometric functions with matrix arguments, which seem to be intractable in themselves. In Ch. 5, The Inference on the Parameters of the Matrix Langevin Distributions, problems of estimation and tests for hypotheses on the parameters are dealt with by Fisher profile scoring methods. These solutions are given in terms of hypergeometric functions with matrix arguments. Bayes estimators, further optimality properties of orientation parameter estimation, and the problem of sufficiency and ancillarity are discussed. In Ch. 6, Large Sample Asymptotic Theorems in Connection with Tests for Uniformity, asymptotic expansions for the density functions of the standardized sample mean matrices and related statistics are derived, and asymptotic properties, near the uniformity, of parameter estimation, and of some optimal tests for uniformity and the asymptotic equivalence of these tests are discussed. Ch. 7, Asymptotic Theorems for Concentrated Matrix Langevin Distributions, is concerned with asymptotic theory for the problem of estimation, sampling distributions, and classification for the concentrated matrix Langevin distributions. The methods are developed to approximately evaluate the estimators of large concentration parameters for each distribution.

Ch. 8, High Dimensional Asymptotic Theorems, investigates the high dimensional asymptotic behavior of matrix statistics and related functions constructed from some main population distributions defined on the special manifolds. The author derives asymptotic expansions for the distributions, generalizes the Stam limit theorems, and discusses asymptotic properties of parameter estimation and tests of hypotheses for the matrix Langevin distributions. Ch. 9, Procrustes Analysis on the Special Manifolds, presents theoretical results obtained by applying Procrustes methods to some statistical analyses on the special manifolds. Procrustes representations of the manifolds and related Procrustes statistics and means are discussed by ordinary, weighted, and generalized Procrustes methods. A brief discussion is given of the isometric and equivariant embeddings of the Stiefel and Grassmann manifolds as spaces defined by the Procrustes representations in Euclidean spaces.

Ch. 10, Density Estimation on the Special Manifolds, develops the theory of density estimation on the special manifolds, where some of the decomposition theorems derived in Ch. 3 play useful roles for the derivations. The theory of density estimation on real spaces of symmetric matrices and of rectangular matrices is developed. Ch. 11, Measures of Orthogonal Association on the Special Manifolds, is concerned with the measurement of orthogonal association on the manifolds, corresponding to linear dependence on the Euclidean space. Population measures of orthogonal association are defined. In particular, on the Stiefel manifold, the author examines the measures of orthogonal association for a family of associated joint distributions, and investigates the asymptotic behavior of the sample measure of orthogonal association for concentrated matrix Langevin distributions. The related problem of orientational regressions is also discussed.

In Appendix A, Invariant Polynomials with Matrix Arguments, a survey of the theory of zonal polynomials (with one matrix argument), including hypergeometric functions with matrix arguments and invariant polynomials with multiple matrix arguments, is presented in consideration of the applications in the text. Appendix B, Generalized Hermite and Laguerre Polynomials with Matrix Arguments, introduces generalized Hermite polynomials and generalized (noncentral) Laguerre polynomials with multiple matrix arguments and discusses various properties of these polynomials, which are useful for the derivations throughout the book. In Appendix C, Edgeworth and Saddle-Point Expansions for Random Matrices, the author develops the methods for obtaining asymptotic expansions for density functions of sample means for random matrices, which are of the Edgeworth, saddle-point, and generalized Edgeworth types, extending the methods for scalar and vector variates.

This book is designed as a reference book for both theoretical and applied statisticians. The book can also be used as a textbook for a graduate course in multivariate analysis for students who specialize in mathematical statistics or multivariate analysis.

Reviewer: Serguey M.Pokas (Odessa)

### MSC:

62Hxx | Multivariate analysis |

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

62E20 | Asymptotic distribution theory in statistics |

33C90 | Applications of hypergeometric functions |

53A99 | Classical differential geometry |

53B99 | Local differential geometry |

### Keywords:

Stiefel manifolds; Grassmann manifolds; orientation statistics; multivariate distributions; distributions of orientations of random matrices; simulation methods for generating pseudo-random matrices; decompositions of special manifolds; sampling; canonical correlation coefficients of general dimension; inference on parameters of matrix Langevin distribudistributions; Fisher scoring methods; maximum likelihood estimators; profile likelihood method; inference on orientation parameters; Bayes estimators; large sample asymptotic theorems; tests for uniformity; asymptotic expansions for sample mean matrices; testing problems; Rayleigh-style tests; likelihood ratio test; Rao score test; profile score functions and tests; locally best invariant tests; concentrated matrices; large concentration parameters; generalized Stam limit theorems; Procrustes analysis; perturbation theory; embeddings; density estimation; measures of orthogonal association; invariant polynomials with matrix arguments; hypergeometric functions with matrix arguments; generalized Hermite and Laguerre polynomials with matrix arguments; Edgeworth expansions for multiple random symmetric metric matrices; random rectangular matrices; generalized multivariate Meixner classes of invariant distributions of multiple random matrices; Edgeworth and saddle-point expansions for random matrices
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\textit{Y. Chikuse}, Statistics on special manifolds. New York, NY: Springer (2003; Zbl 1026.62051)