One has M>p multiple measurments of a p-dimensional random vector.
The objective is to build up, from these M measurements, an
estimate S of the "true" covariance matrix C.
This sample covariance is intended to be used in the standard
role in the "hotelling" quadratic form
D = X'*inv(S)*X
in order to make standard inferential tests on subsequent p-dimensional
random vectors X, (drawn from the putatively stable process from
which the sample covariance was estimated), subject to the
standard assumptions.
For standard inference on D to make any sense, D must be non-negative,
for all X.
Question is:
I am concerned about the positive definiteness of the form D
being robust with respect to sampling.
1.) Does positive definiteness or non-negative definiteness of
the sample matrices S fluctuate (appear and disappear) at random
from random sample to random sample? To put it another way,
is the probability of a zero or negative eigenvalue in a
random sample covariance matrix drawn from a normal population
negligible or not-uncommon?
2.) If one is forced to estimate S from the data on hand, how does
one optimally use the data to estimate a non-negative definite S?
3.) What do you do if you come up with a defective S?
A.) Do you "adjust" the eigenvalues of S?
B.) Do you "adjust" the data and recompute S?
C.) Do you wait for more data before computing S?
D.) What if (C) is not allowed?
Thanks -
Ron Bloom
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
