They were the sons of those Titans who had fought against Jupiter and been sent in chains to the great prison-house of the lower world, but for some reason had escaped punishment. Prometheus, however, did not care for idle life among the gods on Mount Olympus.

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Some cottictit Iliac appcars i t i piint may not be available in electronic format. For inlimnation ahout Wiley products, visit our web site at www. It is designed as a Mr zia il ahq for graduate level courses in multivariate statistical analysis, and I hope that it may also prove to be useful as a reference book for research workers interested in this area.

Any person writing a book in multivariate analysis owes a great debt to T. Anderson for his text, An Introduction 10 Multivariate Statistical Analysis, which has become a classic in the field. This book synthesized various subareas for the first time in a broad overview of the subject and has influenced the direction of recent and current research in theoretical multivariate analysis.

It is also largely responsible for the popularity of many of the multivariate techniques and procedures in common use today.

The current work builds on the foundation laid by Anderson in and in large part is intended to describe some of the developments that have taken place since then. One of the major developments has been the introduction of zonal polynomials and hypergeometric functions of matrix argument by A.

To a very large extent these have made possible a unified Mr zia il ahq of the noncentral distributions that arise in multivariate analysis under the standard assumptions of normal sampling. This work is intended to provide an introduction to some of this theory.

The main focus of this work is on distribution theory, both exact and asymptotic. Multivariate techniques depend heavily on latent roots of random matrices; all of the important latent root distributions are introduced and approximations to them are discussed.

In testing problems the primary emphasis here is on likelihood ratio tests and the distributions of likelihood ratio test statistics, The noncentral distributions vii viii Prejulte are needed to evaluate power functions.

Wherever possible the results of such power studies in the literature are discussed. I I should be mentioned, however, that although the emphasis is on likelihood ratio statistics, many of the techniques introduced here for studying and approximating their distributions can be applied to other test statistics as well.

A few words should be said about the material covered i n the text. Matrix theory is used extensively, and matrix factorizations are extremely important. Most of the relevant material is reviewed in the Appcndix, but some results also appear in the text and as exercises.

Chapter I introduces the multivariate normal distribution and studies its properties, and also provides an introduction to spherical and elliptical distributions. These form an important class of non-normal distributions which have found increasing use in robustness studies where the aim is to determine how sensitive existing multivariate techniques are to multivariate normality assumptions.

In Chapter 2 many of the Jacobians of transformations used in the text are derived, aiid a brief introduction to invariant measures via exterior differential forms is given. A review of rnatrix Kronecker or direct products is also included here, The reason this is given at this point rather than in the Appendix is that very few of the students that I have had i n multivariate analysis courses have been familiar with this product, which is widely used in later work.

Chapter 3 deals with the Wishart and multivariate beta distributions and their properties. Chapter 4, on decision-theoretic estimation of the parameters of a multivariate normal distribution, is rather an anomaly. I would have preferred to incorporate this topic in one of the other chapters, but there seemed to be no natural place for it.

The niaterial here is intended only as an introduction and certainly not as a review of the current state of the art. Indeed, only admissibility or rather, inadmissibility results are presented, and no mention is even made of Bayes procedures. Chapter 5 deals with ordinary, multiple, and partial correlation coefficients.

An introduction to invariance theory and invariant tests is given in Chapter 6. It may be wondered why this topic is included here i n view of the coverage of the relevant basic material in the books by E.

A Decision Theoretic Approach. The answer is that most of the students that have taken my multivariate analysis courses have been unfamiliar with invariance arguments, although they usually meet them in subsequent courses.

For this reason I have long felt that an introduction to invariant tests in a multivariate text would certainly not be out of place. Preluce ix Chapter 7 is where this book departs most significantly from others on multivariate statistical theory.

Here the groundwork is laid for studying the noncentral distribution theory needed in subsequent chapters, where the emphasis is on testing problems in standard multivariate procedures. Zonal polynomials and hypergeometric functions of matrix argument are introduced, and many of their properties needed in later work are derived.

Chapter 8 examines properties, and central and noncentral distributions, of likelihood ratio statistics used for testing standard hypotheses about covariance matrices and mean vectors. An attempt is also made here to explain what happens if these tests are used and the underlying distribution is non-normal.

Chapter 9 deals with the procedure known as principal components, where much attention is focused on the latent roots of the sample covariance matrix.

Asymptotic distributions of these roots are obtained and are used in various inference problems.record length in bytes = number of header records = 6number of samples per record = number of lines in image = number of bytes per sample = 1jpl aircraft sar processor version data type.

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General Muhammad Zia-ul-Haq | Former Army Chief & President of Pakistan