mizhael
Posts:
139
Registered:
12/11/05


Maximum Likelihood Estimation not reliable???
Posted:
Jun 27, 2006 12:45 AM


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 loglikelihood function. And I plug in the simulated data, and then feed this into a wellknown 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 loglikelihood 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 loglikelihood function. Even in a neighborhood of true values, we can still find point that has even larger loglikelihood 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 loglikelihood 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 loglikelihood function might be wrong... I believe the local search part SNOPT should be no problem, because it is just a search.

Thanks a lot!

