Homework 24, Due Friday, May 2, 2008.
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Access the data "bank2.dat" as a text file within the 
directory  http://www.math.umd.edu/~evs/s798c/Data/ .

The file consists of a 200x6 array, of which the first 
100 rows correspond to physical measurements on 100 
real Swiss Bank notes, and the last 100 rows correspond 
to the same physical measurements on 100 forged Swiss 
Bank Notes. In this exercise, consider only the 4th 
column, which corresponds to a measurement in mm of 
the bottom margin of the bank-note.

(1) Separately estimate the densities of the 
lower-margin measurements for the real and forged 
banknotes, using local polynomial density estimation 
with degrees 1, 2 , bandwidth = 0.3, within the 
function "locpoly" in the library (package) KernSmooth.
Over-plot your estimates with the kernel-density estimator
"density" (with optimized bandwidth by the Sheather-Jones
plug-in bandwidth, bw = bw.SJ(xvec)), and report the 
bandwidth used.

(2) Can you see any difference between these different
methods of density estimation ?

(3) Use the best method you can think of to estimate 
for a randomly selected pair of real and forged 
bank-note the probability that the lower-margin 
measurement is greater for the forged note. You may give 
more than one estimate, but choose a best estimate and 
provide some reasoning justifying why you think your 
choice is best.