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Topic: Maximum Likelihood Estimation not reliable???
Replies: 22   Last Post: Jul 4, 2006 5:04 PM

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mizhael

Posts: 139
Registered: 12/11/05
Maximum Likelihood Estimation not reliable???
Posted: Jun 27, 2006 12:45 AM
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Hi all,

I am doing some fun experiments in studying Maximum Likelihood estimation.

Let's say I generate a Geometric Brownian motion process using random number
generator and via some transformation. I've made certain this step is
correct. And I know the true values of drift and sigma that I've used. Let
me call them drift_star and sigma_star.

Now I want to estimate the drift and volatility of my generated data, as an
exercise of MLE:

I've derived analytically the log-likelihood function. And I plug in the
simulated data, and then feed this into a well-known local optimizer, SNOPT.
Funny things occured. Let me now describe the situation:

----------------------

I know MLE needs relative large data set. So I make it 10^6 sample points in
my descritization of the Geometric Brownian motion, for time horizon fixed
at T=1. That's to say, divide T by 10^6 and that's the step size.

I did my local optimization search with a starting point which is exact the
true values, since I simulated this data from the true values.

I was hoping the local optimizer -- SNOPT can converge back to true value
again. In this way, I can verify if MLE does really work, and my analytical
derivation of the log-likelihood function is correct.

The result was surprising. The SNOPT does converge to a local optimum, which
is different the starting point -- the true value. The loglikelihood
function at that point is always larger than that of the true value. I did
many runs of this simulation. The local maximum around the true values are
always larger than that of the true value.

My question is: does this mean MLE is not usable, since the true value point
is not actually the maximum likelihood point of the log-likelihood function.
Even in a neighborhood of true values, we can still find point that has even
larger log-likelihood value? In this case, suppose when we don't know the
true values, then even we take effort to find the global maximum of the
log-likelihood function, it still might not be the true values... not to say
for a hundred data points(not 10^6) in reality.

I hope the more experienced people can help me. If MLE is reliable(given the
sample size is 10^6 which is so big, MLE should converge to true values
asympototically), then my analytical expression of the log-likelihood
function might be wrong... I believe the local search part SNOPT should be
no problem, because it is just a search.

-------------------------

Thanks a lot!





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