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Estimating the covariance matrix K *
Having obtained the estimator of the mean form matrix M described
above, it is fairly simple to obtain the estimator of the covariance K * .
Notice that Y
( LX )( LX ) T ~Wishart (( LM )( LM ) T , L
K L T ) with D degrees of
freedom.
The mean of the non-central Wishart distribution is given by
(Arnold, 1981) E ( Y )
( LM )( LM ) T
K L T ) . Hence, the moment estima-
D ( L
tor of K *
K L T is given by:
L
where M is the estimator of the mean form as obtained at the end of
Step 4.
The estimator of K * obtained above, although square and symmet-
ric, is not guaranteed to be positive semi-definite. One can obtain a
positive semi-definite version using a procedure, sometimes known as
Principal Coordinate Analysis. Consider the spectral decomposition of
the matrix
K * , namely, ˆ
ˆ
PDP T where matrix D is a diagonal
matrix with the diagonal elements corresponding to the eigenvalues of
K * . Replace the negative elements in D by zero and call this modified
matrix ~ D . Obtain a new matrix ˆ
K *
P ~ DP T . This matrix is guaranteed
to be square, symmetric, and positive semi-definite.
The estimator given above is slightly different than the one
described in Lele (1993). In that paper, instead of L
K *
K L T , an estimator
for H
K H T was provided. In this monograph, we use the centering
matrix L , instead of H , to remain consistent with the rest of the chap-
ter. Replacing L in the above description by H retrieves the formulae
and description in Lele (1993).
In the above discussion, we assumed that D
I . This imposes some
restrictions on the applicability of this model. We now consider a more
general situation where the covariance structure is given by K D .
For notational simplicity, let Y
LX ( LX ) T . Let Y
[ Y lm ] where
T ( X )
1,2,…, K denoting the individual elements of the
matrix Y . From the previous section, we know that Y is distributed as
a linear combination of non-central Wishart random variables with
parameters given by (
l
1,2,…, K ; m
K L T ,
D D ) .
Let
D
L
1 1 ,
2 2 ,…,
K *
lm j ] . It follows from their definition that the
matrices j 's are symmetric and that
lm ] and j
[
[
d l j = d l j
j
.
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