ST750 -- Fall 2010

  Homework #6 -- due Thursday, 04 November 2010
  
  Exercises in Textbook: (*just turn in these)
   10.2, 10.8

  New Exercises:

  *1) Repeat Exercise 10.2 by generating from the exponential
    distribution.  (Rewrite the integral so that it looks like
    the expectation of some function of an exponential random 
    variable.)  Estimate the variance of your estimate, and
    determine the sample size needed to get the error below
    1e-5. 

  *2) Recall the Poisson regression problem from Homework #5, 
    with data in rbmites.dat where the first 
    column is y, and use just column 2 (X1) as the only covariate
    (include an intercept).  Mimic chex102s.r or chex102g.r
    to find the posterior mean vector and covariance matrix
    with a flat prior.  (or chex102n.r)

  *3) The Ramus height data are jawbone measurements on 20 boys
    at ages 8, 8.5, 9, 9.5.  Fit a mixed model with a random 
    intercept so that each boy follows the normal model
       Y ~ Normal_4( Xb, V)
    where X has an intercept column and an age column (you can
    recode as you wish), and the covariance matrix V is
      Cov(Y) = V = sigma^2*I_4 + gamma^2*ONE*ONE'
    where ONE is a vector of 4 ones.

    Find the maximum likelihood estimates for b, sigma^2, and
    gamma^2, and report standard errors based on asymptotics.

    Since the two variance parameters are supposed to be positive,
    I'll suggest that you transform using log/exp to allow the
    search to be unbounded.