ST750 -- Fall 2010

  Homework #4 -- due Thursday, 14 October 2010 -- corrected 05 Oct
   
  Exercises in Textbook: (*just turn in these)
   6.15* (use rnorm(howmany) to generate N(0,1); generate dim*n 
    matrix, write function with vector argument and use apply)
   6.25*(modified) Compute the SVD of X and X+E for 3 different sizes
    of 'small' and see if change in singular values matches the results
    of (6.7.4).  Use the US population data again for X, but divide
    the second column by 2000 and the third by 4000000.  Also,the first
    column of E should be all zero.

   8.3, 8.7, 8.8, 8.9, 8.13 (and compute estimate using x below),
    x = (4.82, 4.88, 4.70, 4.72, 4.88, 4.66, 4.94, 4.80)' 
   and 8.22* (use root finder)

   *1*) Another intermediate between the median and the mean is the
   estimator attributed to Huber, which minimizes
     f(mu) = sum( rho(Xi-mu) )
                   / u^2/2        for | u | < c
   where rho(u) = {  
                   \ c|u|-c^2/2    for | u | > c
   a) Find f'(mu) and discuss its properties and computation.
   b) Write a function taking c and a vector of data Xi as arguments 
   and return a function to compute either f(mu) or f'(mu).
   (See 'finder.r' for examples 'dllikex' and 'llikex'.)
   c) Using c=1/2 and the 8 data values above, find the value of mu 
   that either minimizes f(mu) or is a root of f'(mu)=0.