Homework Problem 9, STAT 705, Fall 2015.

Assigned 10/12/2015,  Due Monday 10/19/2015


(1) Generate 1000 iid random variable values  X_1, ..., X_1000 from the 
distribution  Beta(a,b)  with  a=2.2 and  b = 1.5.


(2) Write a function to maximize the likelihood from such a dataset of size n, 
in one of three ways. Make this a user-supplied option for your function. (You can 
make the first option the default, for simplicity in the 3rd part of the problem.)

The first option: with the value  a  unknown  and  b=b0 input and treated as known.

The second option: with the value  a=a0 input and treated as known, but b unknown.

The third option: with both (a,b) parameters treated as unknown.

NOTE that for most optimization methods, the unknown parameter or parameters 
must be given starting values, so it makes sense to provide input values (a,b) 
for all three options.


(3) Apply your function to the full dataset using option 1 only (inputting b=1.5), 
and make sure it gives reasonable results not only for the estimate ahat of  a  
but also for the estimate of the standard deviation of  ahat. (This means that 
you should calculate the "Fisher Information" for the unknown parameter  a .

    Next apply your function successively to 500 subsets of size 80 chosen 
randomly from your full dataset, according to option 1 (only  a  unknown, with 
b  fixed at the true value 1.5). For each application of your function to a 
subset-sample of size 80, you should generate an estimated  a-value and and 
estimated standard error for your estimator. For these 500 runs of your function, 
contrast the theoretical estimates of standard deviation ahat.SE of your a_hat 
estimators (MLEs) coming from the likelihood maximization with the empirical 
variance (sample variance of your 500 a_hat's). They should be reasonably close.