Tentative topics, subject to change.


Intro of Software, why not SAS (for MS student)
Difference to MA622/522 (matrix) or CompSci. (C language)
(Algorithm, problem, Newton iteration with error,
Bootstrap, random number, variable selection).
 

LaTeX. + WinEd or Winshell
not MSword. (0.5 week)


Optimization: (2 weeks)

        finding MLE, (or least squares, M-estimator.)

        finding solution to equations. solving equations

          in one dimension it is easier, using bi-section, R uniroot(),
          You can also write your own Newton( ).
          In multi dimension it is hard, can we get a multi root?
           when function is not monotone, not continuous.

         Numerical derivative: R package numDeriv. function jacobian() etc.

              function  nlm( ) and optim( ) 
              function  maxNR( ) inside package micEcon. (also inside package maxLik?)
              http://www.econ.upenn.edu/~clausen/computing/bfgs.zip
              trust( ) inside the package trust


         example/project: Cox model with partial likelihood or full likelihood.
                          Rank estimator in AFT models with smoothing.
                          (predictable-ness of the function is gone, but OK?)

           iteration with damping...
           in Newton step, be careful inverting a matrix...
           SVD of a matrix
         
        Intro about convex function. 
        find max/min. under constraint. Lagrange multiplier

        Newton-Raphson and alike. 
        SQP, iterative re-weighted least squares. 
                           (solving logistic regression MLE)

R - package programming, interface to C.  (1 week)

Stochastic Optimization: (1 week)

        Robbins-Monro, Kiefer-Wolfowitz

        (this is Jack Wolfowitz, father of Paul Wolfowitz)

E-M algorithm and alike.  (2 weeks)

        parametric EM. 
        nonparametric EM example. (stochastic approx. use with MC)
        Turnbull EM, Kaplan-Meier estimator.
        Example of Buckley-James estimator.

Re-sampling method: (3.5 weeks)   

        (bootstrap, Jackknife, etc) 
        Second order accuracy example. Some expansion proof

Monte Carlo: (1 week)
        
       r.v. generation. Rejection method.
       Compute probability or expectation. Variance reduction.

MCMC  theory and practice: (2 weeks)
       R package mcmc 


(if time permit)
Cross validation
Kernel smoothing
Bartlet Correction example

======================================================


Projects: 2009 

General comments: There do not need to have new theoratical component
in your project. (if you have, it is certainly good to have, but not required)
But there do need to have a substantial computational part and your effort.
(required)



Speeding up existing R package. (Em type iteration)

Two sample Empirical Likelihood mean diff



Past projects
==============

Selection of best fitting logistic regression model to the microarray data.
Strategy of the search. criterior of a good model: 
AUC, log likelihoods and others. (are they related? given similar results?
which one is easier to use? convex?)

effort to speed up computation? time comparison?



Alternative Residual sum of squares calculation in 
 the Buckley-James estimation method and its consequences 
(affecting model selection? prediction error? or other M-estimators).
Can this be used to estimate variance? R-square?)



fitting  the polynomials * weight functions 
to the general regression function, search for the best tuning parameters.
(how to search: convexity?) computational time?


Using Dirichlet Process prior and Gibbs sampler for density estimation
Want to implement own version of Gibbs sampler. (dimension = 1000 ?)
(how is that different to the finite dimensional approach ?)

How dimensional increase slow you down? memory consumption?



Smoothed estimating function in the Rank Regression, and its 
solver (need R coding).
Compare to unsmoothed version (in terms of computation, the final estimators).

computational time?  memory usage?