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# Multivariate

### Overview

This course aims to teach students about multivariate statistical analysis. By the end of the course, learners will be able to conduct multivariate tests, calculate probabilities for bivariate normal random variables, perform linear discriminant analysis, understand the distribution of quadratic forms, and apply principal components analysis. The course covers topics such as statistical distance, multivariate descriptive statistics, multivariate normal distribution, and testing hypotheses using Follmann's test. The teaching method involves using R for practical applications. This course is intended for individuals interested in expanding their knowledge of advanced statistical techniques and their applications in various fields.

### Syllabus

A Simple Multivariate Test for One Sided Alternatives.
The Multivariate Sign Test.
Probability of 1st Quadrant for a Scaled Bivariate Normal Random Variable.
Using R: Calculating Probability for a Bivariate Normal Random Variable.
Using R: The Multivariate Sign Test.
Power and Sample Size in R: Multivariate Sign Test.
Statistical Distance.
A Square-Root Matrix.
Extended Cauchy-Schwarz Inequality.
Linear Discriminant Analysis.
Distribution of Quadratic Forms (part 1).
Distribution of Quadratic Forms (part 2).
Distribution of Quadratic Forms (part 3).
Gaussian Integrals.
Spherical Coordinates.
Multivariate Normal Random Variable transformed to a Multivariate Uniform Random Variable.
Rotational Invariance.
(1-a)% Confidence Region for a multivariate mean vector when the data are multivariate normal.
Multivariate Descriptive Statistics.
Multivariate Normal Distribution as an approximation to the Multinomial Distribution.
Testing all Treatments Arms against a Control Arm using Follmann's Test.
Dose Escalation Hypotheses Testing using Follmann's Test.
Using R to test multivariate ordered alternatives with Follmann's test.
Cov(y1, y2)=0 if and only if (y1 independent of y2).
Random Vectors and Random Matrices.
Principal Components (part 1): Background.
Principal Components (part 2): Derivation.
Principal Components (part 3): "Explained" Variance.
Principal Components (part 4): Correlation.

statisticsmatt