## A First Course in Multivariate StatisticsMy 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. |

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Really comprehensive and well presented. shows the beauty of statistics.

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great book. Straight to the point and Very helpful!

### Contents

Why Multivariate Statistics? | 1 |

Exercises for Chapter 1 | 18 |

Joint Distribution of Several Random Variables | 23 |

22 Probability Density Function and Distribution Function of a Bivariate Random Variable | 25 |

23 Marginal Distributions | 38 |

24 Independence of Random Variables | 47 |

25 Expected Values Moments Covariance and Correlation | 57 |

26 Conditional Distributions | 75 |

66 UnionIntersection and Likelihood Ratio Testing | 418 |

67 ResamplingBased Testing | 435 |

Discrimination and Classification Round 2 | 453 |

Linear vs Quadratic | 460 |

73 Canonical Discriminant Functions | 485 |

74 Multivariate Analysis of Variance | 509 |

75 Simple Logistic Regression | 519 |

76 Multiple Logistic Regression | 538 |

27 Conditional Expectation and Regression | 89 |

28 Mixed DiscreteContinuous Distributions and Finite Mixtures | 104 |

29 Sums of Random Variables | 130 |

210 Notions and Concepts of pvariate Distributions | 140 |

211 Transformations of Random Vectors | 155 |

The Multivariate Normal Distribution | 171 |

32 Definition and Properties of the Multivariate Normal Distribution | 175 |

33 Further Properties of the Multivariate Normal Distribution | 186 |

34 Spherical and Elliptical Distributions | 197 |

Parameter Estimation | 209 |

42 Plugin Estimators | 216 |

43 Maximum Likelihood Estimation | 233 |

44 Maximum Likelihood Estimation with Incomplete Data | 260 |

Discrimination and Classification Round 1 | 279 |

52 Standard Distance and the Linear Discriminant Function | 280 |

53 Using the Linear Discriminant Function | 302 |

54 Normal Theory Linear Discrimination | 323 |

55 Error Rates | 344 |

56 Linear Discriminant Functions and Conditional Means | 357 |

Statistical Inference for Means | 375 |

62 The OneSample PTest | 377 |

63 Confidence Regions for Mean Vectors | 391 |

64 The TwoSample Ptest | 402 |

65 Inference for Discriminant Function Coefficients | 408 |

Linear Principal Component Analysis | 563 |

82 SelfConsistent Approximations | 568 |

83 SelfConsistent Projections and Orthogonal Least Squares | 578 |

84 Properties of Linear Principal Components | 592 |

85 Applications | 605 |

86 Sampling Properties | 617 |

87 Outlook | 625 |

Normal Mixtures | 639 |

92 Maximum Likelihood Estimation | 645 |

93 The EMAlgorithm for Normal Mixtures | 656 |

94 Examples | 663 |

Normal Theory Discrimination with Partially Classified Data | 679 |

Selected Results From Matrix Algebra | 687 |

A1 Partitioned Matrices | 688 |

A2 Positive Definite Matrices | 689 |

A3 The Cholesky Decomposition | 690 |

A4 Vector and Matrix Differentiation | 691 |

A5 Eigenvectors and Eigenvalues | 692 |

A6 Spectral Decomposition of Symmetric Matrices | 693 |

A7 The Square Root of a Positive Definite Symmetric Matrix | 695 |

Bibliography | 703 |

711 | |

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### Common terms and phrases

assume bivariate normal bivariate random chi-square distribution classification regions classification rule compute conditional distribution conditional mean consider continuation of Example correlation Cov[X defined degrees of freedom denote density function dimension eigenvalues eigenvectors equal error rate Exercises for Section Figure follows fxr(x,y given graph Hint hypothesis independent interval joint distribution joint pdf likelihood function linear combination linear discriminant function log-likelihood function log-likelihood ratio log-likelihood ratio statistic logistic regression marginal distribution maximum likelihood estimates mean vector midge mixture density multivariate normal distribution multivariate standard distance nonsingular normal mixture normal theory observed data obtained optimal classification orthogonal otherwise p-variate random vector parameter estimates partitioned positive definite posterior probabilities Pr[X principal component approximation prior probabilities probability function proof Prove equation quadratic random vector redundancy result sample covariance matrix scatterplot self-consistent setup Show standard errors Suppose symmetric Table Theorem transformation univariate variance wing length zero