DISCUSSION FOR ASSIGNMENT 2


PROBLEM 2:  Estimation of mu with rho fixed at log(0.5).

I tried starting points of mu = 0.0, 2.5, 5.0 for each of the three
data sets.

For data set 1, N-R rapidly converged to mu=1.38 from all three
starting points.  Looking for some other behaviour, I tried starting
near the opposite side of the circle, at mu=4.5.  This produced a
value of mu=4.498, which appears to be a local minimum, rather than a
maximum.

For data set 2, I found two local maxima, at mu=5.179 and mu=2.037,
with mu=5.179 being the better of the two.

For data set 3, convergence to two points was also found, with
mu=1.453 being a maximum and mu=4.779 being a minimum.


PROBLEM 3:  Estimation of mu and rho jointly.

I again tried starting at mu = 0.0, 2.5, 5.0 with rho = log(0.5).

For data set 1, none of these starting points worked - they all
converged very slowly, to a point where the estimated covariance
matrix was not positive definite (as seen by one the standard errors
being NaN).  I tried starting closer to the solution from Problem 2,
with mu = 1.3 and rho = log(0.1).  This also didn't work.  Only by
starting very close to the solution, with mu=1.35 and rho=log(0.1),
did I finally get convergence.

For data set 2, two of the starting points converged to what seems to
be the global maximum, at mu=5.18 and rho=0.455.  I tried to get
convergence to the other local maximum found in Problem 2 by starting
at mu=2.0, but it diverged to something other than a local maximum
instead.  (This local maximum may not exist when rho is not fixed.)

For data set 3, one of the starting points converged to what appears
to be the global maximum, at mu=1.488 and rho=0.409.  From other
starting points, N-R moved slowly to some point that isn't a local
maximum.