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.