On page 245, lines 3 and 4, of the published paper, we find the following text:
“Since
$\mbox {tr}\, \{U_d \Gamma \} $
can be expressed as the trace of the product of two positive definite matrices,
$\mbox {tr}\, \{U_d \Gamma \}>0$
, and thus
$c_d>0$
;”
This text should be replaced with:
“Since
$ \mbox {tr}\, \{ U_d \Gamma \} $
can be expressed as the trace of a positive definite matrix,
$\mbox {tr}\, \{U_d \Gamma \}>0$
, and thus
$c_d>0$
;”
The uncorrected text claims that
$U_d$
and
$\Gamma $
are positive definite matrices, but
$U_d$
can’t be positive definite, since its rank (difference between the ranks of the derivatives of the two models involved) is much less than its order.
The expression
$\mbox {tr}\, \{ U_d \Gamma \} $
could be written differently so that the conclusion still holds. Namely, write
$U_d = V \Pi P^{-1} A^\prime (A P^{-1} A^\prime )^{-1} A P^{-1} \Pi ^\prime V $
(formula (4) of the paper) as
$U_d = FF^\prime $
, where
${F =V \Pi P^{-1} A^\prime (A P^{-1} A^\prime )^{-1/2}}$
; then,
$\mbox {tr}\, \{ U_d \Gamma \} = \mbox {tr}\, \{ FF^\prime \Gamma \} = \mbox {tr}\, \{ F^\prime \Gamma F \},$
where
$F^\prime \Gamma F$
is a positive definite matrix, given that
$\Gamma $
is positive definite in the setup of the paper.
For rewriting the alternative expression of
$\mbox {tr}\, \{ U_d \Gamma \}$
, we used the well-known matrix algebra result that
$\mbox {tr}\, \{M N \} =\mbox {tr}\, \{N M \} $
for matrices M and N of dimensions conformable with the products; in our application,
$M= F$
and
$N=F^{\prime } \Gamma $
.
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