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Covariance matrix estimation of timedependent distributions
Posted:
Jun 17, 1996 7:04 AM


Hi, distribution F. Let there be a sequence of realisations x(t), t = 1,2,3 ... of X at discrete time steps t. I wish to estimate the covariance matrix of X but:  The distribution F is changing 'slowly' with time.
In case of constant distribution estimators (:= C(n) at time t=n ) for the covariance matrix are: 1 n _ _ T _ 1 n C(n) :=  \sum (x(t)  x) (x(t)  x) , x :=  \sum x(t) n t=1 n t=1
or: 1 ...  ... n1
 If I have C(n), I don't want to store the values for x(t), t < n anymore.
Problem: The distribution F of X changes with time and if I estimate with C(n) at time t=n, there is the problem that initial values of x(t), t << n, have been carried forward with every subsequent calculation and they disturb the value of the estimation C(n) of the covariance matrix of X with actual distribution F.
Questions: i) Are there any theoretic results about this problem ?  the relationship between the changing distribution and the value of the estimated covariance matrix ?  how quick/good does C(n) converge ?
ii) How many realisations of X do I need for a 'good' estmation for the covariance matrix (in general) ?
iii) Is there another estimator for the covariance matrix ?
iv) A possibility to estimate the covariance matrix incrementally, meaning to forget the older values of x(t)? I'm interested in practical aid as well as in theoretical results. Thank you very much.
Peter Schnaus dipedro@informatik.unimainz.de Uni Mainz, Germany



