To all who have helped me on the previous thread thank you very much. I am
reposting this beause the question has become more focused.
I am studying a stochastic Markov process and using a maximum likelihood
technique to fit observed data to theoretical models. As a first step I am
using a Monte Carlo technique to generate simulated data from a known model to
see if my fitting method is acurate. In particular I want to know if I can use
this techniques to dtermine the number of free parameters in the Markov Model.
I have been using the Log(Likelihood) method which seems to be widely acceted.
I am getting very small Log(Likelihood ratios) in cases when I know the more
complex model is correct (i.e. H0 should be rejected). When I first observed
this I tried increasing the N values, and found a decrease rather than an
increase in the Log(Likelihood ratio). I now think I know why. I am posting in
hopes of finding out if my proposed solution is 1)statistical heracy, 2)so
obvious that I should have realized it 6 months ago, or 3)a plausible idea in
need of validation.
The likelihood fuction I have been using up to now which I will call the FULL
likelihood function is:
L= (1/Sqrt( | CV-Matrix |))*exp((-1/2)*(O-E).(CV-Matrix^-1).(O-E))
Where | CV-Matrix | is the determinant of the Covariance matrix, (O) is the
vector of observed values in time order and (E) is the vector of the values
predicted by the Markov model for the corresponding times. The Covariance
matrix is generated by the Markov model.
IN A NUTSHELL: It appears that the factor (1/Sqrt( | CV-Matrix |)) is the
source of the problem. In many MLE discriptions this is a constant and drops
out. In my case there is a big difference between the (1/Sqrt( | CV-Matrix |))
for different models (several log units). I believe this may be biasing the fit
in some way.
MY PROPOSAL: I have begun fitting my data to the follwing simplified likelihood
formula:
L= exp((-1/2)*(O-E).(CV-Matrix^-1).(O-E)).
Does this seem reasonable?
Thanks for any insight
James Celentano
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