Homework Problem 11, Due Monday March 3.
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(This due-date has been extended because of the correction
inserted below changing exp(-X) to exp(-|X|), on Feb.28.)

(a) Generate 1000 independent batches of 
20 i.i.d. random 2-vectors (X_i,Y_i), i=1,..,20 
for which

X has a standard logistic distribution, and 
      conditionally given X

Y - X ~ N( 0, 2*(1+exp(-|X|)) )   [CORRECTION HAS BEEN INSERTED HERE]

Give some distributional checks or graphical 
evidence to show persuasively that you have 
simulated the joint (X,Y) distribution correctly.


(b) The objective of this simulation is to evaluate

E( (Y_(10) - X_(10))^2 )               (*)

where the subscript _(10) refers to the 10th order 
statistic (ie the 10th smallest observation) within 
the batch.

One method of evaluating the method (*) is by a 
direct simulation: perform (and time!) a simulation 
of 10000 batches and find (*) as a direct empirical 
average of 10000 quantities  (Y_(10) - X_(10))^2  that
you simulate.

Explore whether you can speed up the simulation (while 
obtaining accuracy at least as good as with your first
method) by using the method of control variates with 
control variate  2*(1+exp(-|X_(10)|)).

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NOTE: it is not possible to get much of a speedup here.
(Actually, the problem statement was feasible with 
variance of the form 2*(1+exp(-X)), and the speedup 
that way was marginally better than in the current 
formulation.)

But I will show in class that the same method on a 
betterconstructed problem gives a dramatic speedup. 
The modified problem is:

X ~ logistic as before, and conditionally given X,

Y  = 5 * X *( 1 + Unif(.5,1.5) )

Problem is again to find  E( (Y_(10)- X_(10))^2 )
based on order-statistics from datasets of size 20.