ST790R -- Fall 2009

  Homework #6 -- due Thursday, 05 November 2009
 
  Exercises in Textbook: (*just turn in these)
   10.2, *10.6(just 3 points), 10.8

  New Exercises:

  *1) Repeat Exercise 10.2 by generating from the exponential
    distribution.  For a modest sized sample, estimate the variance 
    of your estimate of the integral, and determine the sample size 
    (N) needed to get the error below 1e-5. (No need to generate
    a sample of size N; no need for a tail bound t.)

  2) Recall the logistic regression Example 9.7 in chex97.r/out
    with one covariate.  Mimic chex102s.r or chex102g.r
    to find the posterior mean vector and covariance matrix
    with a flat prior.  (or chex102n.r)

  *3) Many NC health statistics are reported as county rates, that
    is, cases per 1000 residents.  A reasonable model for the 
    variance for these rates is
       sigma^2 + gamma^2/pop(i)
    where pop(i) is the population of the county.  In the file
    'ncinfmort.ddat' are county population (k), infant death
    rates (deaths/1000), household income ($k), and poverty rate.
    We want to fit a regression model of the form
       infdr(i) = beta0 + beta1*hinc(i) + beta2*pov(i) + e(i)
    where indr(i) is infant death rate, hinc(i) household income,
    and pov(i) poverty rate in county i=1,...,N=100.

    a) Compute least squares estimates for the model above.
    b) Compute preliminary estimates for sigma^2 and gamma^2
    by regressing the square of the residuals on 1/pop(i).
    c) Compute ML estimates of the parameters (3 beta's, two
    variance params) fitting the regression model with the 
    covariance model that depends on population.
    (likelihood from y ~ N(Xb,V), V=diag(sigma^2+gamma^2/pop) )
    (also report standard errors for MLE's)
    d) Do a LRT of H: gamma^2 = 0 at level 0.05.