LinearModels

This repo is based on 250 series at the Biostat Department in UCLA.

Instructors: Dr. Weng Kee Wong and Dr. Sudipto Banerjee.

• Lecture notes are written by Elvis Cui and some materials are from scientific papers.

• Bayesian topics include:

  1. Bayesian conjugate linear models (models with conjugate prior).
  2. Connection between Bayesian and frequentist linear models.
  3. Sampling methods such as composition sampling, method of mixtures, MCMC algorithms, etc.
  4. Brook's lemma (aka Hammersley-Clifford theorem).
  5. Sherman-Woodbury-Morrison formula using multivariate statistics theory.
  6. Sequential Bayesian learning.
  7. Directed acyclic graph (DAG).
  8. Alternative ways to look at normal densities.
  9. Spatial models (CAR).
  10. Tensor product models, aka matrix-regression models.

• Classical topics include:

  1. Matrix theory with an emphasis on projection operators.
  2. Distribution theory, especially Fisher-Cochran's theorem.
  3. Least square theory: constrained estimation, conditions for a parameter to be estimable, etc.
  4. Multiple and partial correlation coefficients.
  5. Violation of assumptions in linear models and remedies.
  6. Hypothesis testing such as omnibus test, Fieller's theorem, Cook-Weiesberg's test, etc.
  7. Simultaneous inference such as Tukey's q, Scheffe's method, etc.
  8. Shrinkage and Bayes estimation (an unusual version of James-Stein estimator and its statistical properties are given).
  9. ANOVA mixed models.
  10. Linear mixed models with REML.

• Course materials are included.

• Homework are included (my own solutions).