Topics for Test Study
===================== 3/9/22
I. Linear Algebra / Matrix Algebra
(mostly from lectures, with MKB Appendix as reference, especially Secs. A.6.2 and A.7)
A. Projections
B. Spectral Decomposition
C. det and trace in terms of eigenvalues
D. Some simple min-max problems for matrices (Sec. A.9)
II. Properties of Multivariate Normal (lecture, MKB sec 2.5, 3-1-3.3)
A. Equivalent definitions (nonsingular case) in
terms of density, ch. function, linear
transformation from vecctor with iid N(0,1) entries
B. Spherical symmetry of N_p(0, sig^2 * I)
C. Conditional distribution of one subvector given another
D. Best linear approximations of normal components within
multivariate normal
E. Independence of Quadratic Forms from matrices B, C
with B C = 0
III. Distributions of Statistics derived from Normal data matrices
A. Wishart Distribution, Hotelling T^2 def'ns and basic properties (MKB Secs. 3.4-3.7)
B. Independence of Ybar and S from normal data matrices
C. Maximum Likelihood for mu and Sigma in one- and multi-sample
normal data matrices (MKB Sec. 4.2)
IV. Likelihood Ratio Tests for basic hypotheses
A. For means: equality or subvector 0 or [proportionality
to given vector]
B. For variances: tests of equality to given Sig0, or
two-sample test of equalityof equality
C. Multivariate Regression -- basic LRTs for B, plus
MANOVA (one-way)
V. Union Intersection Tests --definition and basic examples MKB Sections 5.2.2 through 5.3
giving simultaneous CIs Section 5.5.
VI. Simulation --
The main idea is to rely on the law of large numbers idea to ensure that
distribution functions and quantiles of functions of normal data matrices
(of fixed size n x p) can be estimated to arbitrarily good accuracy by
simulating a large number of replicates R of the nxp matrices. The
application is to get rejection cutoffs for various LRT or UIT test
statistics, and then also to calculate/estimate power for the resulting
test and coverage for the resulting confidence intervals. All of these
simulations can also be applied to large numbers of nxp statistics from
NON-normal data matrices, to check robustness.
One particular simulation idea we discussed in some detail in class was the
relation between spherical symmetry and normal iid variables to give a fasrt cheap
way to simulate random unit vectors in any dimension.