CSI 779, Spring 1995, J. E. Gentle

Assignment due Feb 20

First, prepare a brief write-up about your project:
 A.  Give the full reference for the article you are  
     using.
 B.  In one or two sentences describe the author's  
     objective.
 C.  Identify the measured response(s) in the author's  
     Monte Carlo study.
 D.  Identify the factors and the levels that the  
     author used.
 E.  Define your planned study (responses, levels of  
     factors, etc.)
 F.  Identify what software (and the functions within  
     that software) that you plan to use.
then, either
*put all of this in one or more files accessible from  
  your Mosaic page
or
*email to me a brief ASCII write-up about your project  
  (i.e., no mathematical equations).  This should only  
  be about 30 to 100 lines.  

Second, work the following exercises and email me the solutions.

1.  A student wanted to use a normal distribution with  
    mean of 100 and variance  of 10.  In S-Plus, he generated  
    a random sample of size 500 from such a distribution, and  
    computed a 95% confidence interval for the mean.
    The student then decided to evaluate the random number  
    generator in S-Plus.  He generated 100 samples of size  
    100 from the normal with mean of 100 and variance of 10  
    (as before), and counted how many of these sample means  
    fell within the confidence interval he had computed earlier.
    (He expected to get about 95.)

    Do this yourself in S-Plus.
    Show your program.
    Describe your results.
    Explain your results.


2. Redo (from Assignment a0213) the nonparametric bootstrap  
   estimate of the standard deviation of the correlation  
   coefficient between the two columns of the data in law.dat in  
   my directory.  This time, start with a bootstrap sample of  
   size 10, display your results; increase the sample size to 20  
   (just generate 10 additional observations and use the original  
   10 with them), display your results; and continue this way  
   until your bootstrap sample size is 200.  Plot a histogram  
   ({\tt hist}) of the 200 bootstrap observation.  
   Comment on what you observe (two or three sentences).

3. Consider a bootstrap estimate of the variance of an estimator T.
   Prove that the estimate from a bootstrap sample of size m has the
   same expected value as the ideal bootstrap estimator, but that its
   variance is greater than or equal to that of the ideal bootstrap
   estimate.  (This is the variance of the variance estimator.  Also,
   note that the expectation and variance of these random variables
   should be taken with respect to the true distribution, not the
   empirical distribution.)
   Email to me only an outline of your proofs.

4. Compute a parametric bootstrap estimate of the standard deviation of
   the correlation coefficient by first estimating the parameters of an
   assumed normal distribution.  Use a bootstrap sample of size 200.
   (Use {\tt chol} and {\tt rnorm}.)