Homework 10, Due Friday Oct. 16, 2009.
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Generate a 1000-observation dataset  HW10Data 
using the command:

> HW10Data = rlogis(1000, runif(1,2,2.5), runif(1,.7,.85))

This gives a pseudo-random sample of size 10000 from a 
logistic distribution with location parameter  A
and scale-parameter B, which are unknown parameters,
and which you are to estimate by numerical maximi-
zation of a (misspecified) logistic log-likelihood.

(For the sake of good starting values, you may use the 
fact that the true parameter values fall in the 
interval  (2,2.5) for  A  , and  (.7,.85)  for  B.

Your task in this exercise is to find the Maximum 
Likelihood Estimators and Standard Errors for the 
location parameter  A  and scale-parameter B, 
treating your  data as though they came from a 
logistic density    dlogis((x-A)/B)/B  .

      Do this by setting up a direct numerical 
maximization using  nlm.  First do it without 
providing "gradient" and "hessian" attributes, and 
then do it again providing these attributes coded 
as efficiently as you can. Do "unix.time" timing 
runs both ways, to compare the computing time with 
and without analytical gradient and hessian.


(b) Also do the minimization by the "optim" function, 
and by a direct Gradient search using the same log-likelihood 
function (using your same analytical gradient function), 
and do a "unix.time" timing run on it.

NEEDLESS TO SAY, YOUR DIFFERENT MAXIMIZATION ROUTINES 
SHOULD ALL REACH THE SAME MAXIMUM !!