This course covers background material on general linear models, focusing on topics such as random vectors, statistical distributions, idempotent matrices, and quadratic forms. The course aims to teach students about the theoretical foundations and mathematical concepts essential for understanding regression models. The teaching method involves video lectures that are structured in a sequential order for better comprehension. This course is intended for learners who are already familiar with regression models and want to deepen their understanding of the underlying mathematical principles.
Overview
Syllabus
Random Vectors and Random Matrices.
Statistical Distributions: Central & Noncentral t Distributions.
Statistical Distributions: Central & Noncentral Chi square df=1 Distributions.
Statistical Distributions: Derive the F Distribution.
Statistical Distributions: NonCentral F Distribution.
Idempotent Matrices.
Independence of Quadratic Forms.
Independence of Quadratic Forms (another proof).
Distribution of quadratic form n(xbar-mu)Sigma(xbar-mu), where x~MVN(mu,sigma).
Distribution of Quadratic Forms (part 1).
Distribution of Quadratic Forms (part 2).
Distribution of Quadratic Forms (part 3).
(1-a)% Confidence Region for a multivariate mean vector when the data are multivariate normal.
Derivative of a Quadratic Form with respect to a Vector.
Projection Matrices: Introduction.
Perpendicular Projection Matrix.
Mean, Variance, and Covariance of Quadratic Forms.
A Square-Root Matrix.
Inverse of a Partitioned Matrix.
The Spectral Decomposition (Eigendecomposition).
Woodbury Matrix Identity & Sherman-Morrison Formula.
Generalized Inverse Matrix.
Generalized Inverse for a Symmetric Matrix.
Gram-Schmidt Orthonormalization Process: Perpendicular Projection Matrix.
Sum of Perpendicular Projection Matrices.
Taught by
statisticsmatt
