Homework 11, Assigned 10/16/17, due 10/25, 6pm
==============================================       14 points

(NOTE the due-date is deferred to Wednesday 10/25, although it was 
originally announced as Monday 10/23) 

Consider the dataset of 123 (uncensored) failure-time observations obtained by downloading

exa1 = read.csv("https://raw.githubusercontent.com/CodeOwl94/ross-reliability/master/EXA1.csv",
                 header=T)

and retaining only uncensored equipment failure times

failtim = exa1$time[exa1$fail=="F"]

Common distributions used to describe failure-time data are Gamma and logNormal, both 
2-parameter distributions.

(a) Fit   lognormal(mu,sig)  and   Gamma(a,b) densities to these data by the method of 
maximum likelihood. We refer to the 2-dimensional parameter for these models as theta

(b) For each of the distributions fitted in (a), display the (empirical, or observed) 
Fisher Information matrices A, and compare these with the alternative-form matrices B
given by (1/n) sum_{i=1}^n v_i v_i^{tr}   where   v_i  is the gradient of the log-density 
evaluated at the i'th data-value X_i and the MLE parameter value.

    Do you see an indication of lack of fit of one or both of these models by comparison 
of the A versus B matrices ?

(c) Obtain for each model 
       (i) the variance-covariance matrix of the 2-dimensional theta parameter-estimator 
       that you would report if (you believed that) the model were properly specified,   and

       (ii) the "robustified" or sandwich-formula variance-covariance matrix for the same 
       parameter estimator.

(d) Overplot the histogram for the log-transformed data with the density curves for your fitted 
models (a) (in blue) and (b) (in red), and label your picture appropriately with x 
and y axis labels, a title, and a "legend" showing which line is associated with which model.

(e) Can you draw any conclusions about which density is a better fit for these data, or 
if either is any good ?