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## Über dieses Buch

My goal in writing this book has been to provide teachers and students of multi­ variate statistics with a unified treatment ofboth theoretical and practical aspects of this fascinating area. The text is designed for a broad readership, including advanced undergraduate students and graduate students in statistics, graduate students in bi­ ology, anthropology, life sciences, and other areas, and postgraduate students. The style of this book reflects my beliefthat the common distinction between multivariate statistical theory and multivariate methods is artificial and should be abandoned. I hope that readers who are mostly interested in practical applications will find the theory accessible and interesting. Similarly I hope to show to more mathematically interested students that multivariate statistical modelling is much more than applying formulas to data sets. The text covers mostly parametric models, but gives brief introductions to computer-intensive methods such as the bootstrap and randomization tests as well. The selection of material reflects my own preferences and views. My principle in writing this text has been to restrict the presentation to relatively few topics, but cover these in detail. This should allow the student to study an area deeply enough to feel comfortable with it, and to start reading more advanced books or articles on the same topic.

## Inhaltsverzeichnis

### 1. Why Multivariate Statistics?

Abstract
There is no better way to arouse interest in multivariate statistics than to study some good examples. This chapter gives four examples, three of them based on real data. The fourth one is a brainteaser.
Bernard Flury

### 2. Joint Distribution of Several Random Variables

Abstract
Multivariate statistical methods are useful when they offer some advantage over a variable by variable approach. In Example 1.1, we saw that considering one variable at a time may not be optimal for classification purposes. Hence, it is important that we establish some terminology and acquire a basic knowledge of bivariate and multivariate distribution theory. In particular, we shall discuss notions such as independence of random variables, covariance and correlation, marginal and conditional distributions, and linear transformations of random variables. In the spirit of keeping this chapter on an elementary level, we shall restrict ourselves to the case of two variables most of the time, and outline the general case of p ≥ 2 jointly distributed random variables only toward the end in Section 2.10.
Bernard Flury

### 3. The Multivariate Normal Distribution

Abstract
Before introducing the multivariate normal distribution, let us briefly review some important results about the univariate normal.
Bernard Flury

### 4. Parameter Estimation

Abstract
Estimation of parameters is a central topic in statistics. In probability theory we study the distribution of random variables, assuming they follow certain distributions, and try to find out what is likely to happen and what is unlikely. Conversely, in statistics we observe data and try to find out which distribution generated the data. In the words of my colleague R.B. Fisher: “In probability, God gives us the parameters and we figure out what is going to happen. In statistics, things have already happened, and we are trying to figure out how God set the parameters.”
Bernard Flury

### 5. Discrimination and Classification, Round 1

Abstract
Discriminant analysis and related methods, treated in Chapters 5 to 7, are the central topics of this course. Chapter 5 gives an introduction on a mostly descriptive level, ignoring questions of statistical inference. The mathematical level of Chapter 5 is moderate, and all concepts are explained at great length, hoping that even students without a strong mathematical background will be able to master most of the material. Chapter 6 gives an introduction to problems of statistical inference that arise naturally from the setup of discriminant analysis: testing for equality of mean vectors, confidence regions for mean vectors, and related problems for discriminant functions. Then Chapter 7 resumes the classification theory on a more abstract level and gives brief introductions to related topics, such as logistic regression and multivariate analysis of variance.
Bernard Flury

### 6. Statistical Inference for Means

Abstract
In this chapter we study selected problems of hypothesis testing and confidence regions in multivariate statistics. We will focus mostly on T 2-tests, or Hotelling’s T 2, after the statistician Harold Hotelling (1895–1973). In the spirit of this book, which emphasizes parameter estimation more than testing, we will give rather less attention to aspects of hypotheses testing than traditional textbooks on multivariate statistics. In particular, we will largely ignore problems like optimality criteria or power of tests. Instead, we will focus on a heuristic foundation to the T 2-test methodology, for which we are well prepared from Chapter 5.
Bernard Flury

### 7. Discrimination and Classification, Round 2

Abstract
In this chapter we continue the theory of classification developed in Chapter 5 on a somewhat more general level. We start out with some basic consideration of optimality. In the notation introduced in Section 5.4, Y will denote a p-variate random vector measured in k groups (or populations). Let X denote a discrete random variable that indicates group membership, i.e., takes values 1, … , k. The probabilities
$${\pi _j} = \Pr \left[ {X = j} \right]\quad j = 1, \ldots ,k,$$
(1)
will be referred to as prior probabilities, as usual. Suppose that the distribution of Y in the jth group is given by a pdf f j (y), which may be regarded as the conditional pdf of Y, given X = j. Assume for simplicity that Y is continuous with sample space ℝ p in each group. Then the joint pdf of X and Y, as seen from Sec tion 2.8, is
$${f_{XY}}\left( {j,y} \right) = \left\{ \begin{gathered} {\pi _j}{f_j}\left( y \right)\;for{\kern 1pt} j = 1, \ldots ,k,y \in {\mathbb{R}^p} \hfill \\ 0\quad otherwise. \hfill \\ \end{gathered} \right.$$
(2)
Bernard Flury

### 8. Linear Principal Component Analysis

Abstract
Principal component analysis is a classical multivariate technique dating back to publications by Pearson (1901) and Hotelling (1933). Pearson focused on the aspect of approximation: Given a p-variate random vector (or a “system of points in space,” in Pearson’s terminology), find an optimal approximation in a linear subspace of lower dimension. More specifically, Pearson studied the problem of fitting a line to multivariate data so as to minimize the sum of squared deviations of the points from the line, deviation being measured orthogonally to the line. We will discuss Pearson’s approach in Section 8.3; however, it will be treated in a somewhat more abstract way by studying approximations of multivariate random vectors using the criterion of mean-squared error.
Bernard Flury

### 9. Normal Mixtures

Abstract
This chapter gives a brief introduction to theory and applications of finite mixtures, focusing on the most studied and reasonably well-understood case of normal components. An introductory example has already been given in Chapter 1 (Example 1.3) and has been discussed to some extent in Section 2.8, where the relevant terminology has been established. Before turning to the mathematical setup, let us look at yet another example to motivate the theory.
Bernard Flury

### Backmatter

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