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| ADMISSIBLE ESTIMATES IN THE IMPORTANT CLASS OFESTIMATES OF THE COVARIANCE MATRIX |
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Citation： |
Xie Minyu.ADMISSIBLE ESTIMATES IN THE IMPORTANT CLASS OFESTIMATES OF THE COVARIANCE MATRIX[J].Chinese Annals of Mathematics B,1994,15(4):435~442 |
| Page view： 1016
Net amount： 799 |
Authors： |
Xie Minyu; |
Foundation： |
Project supported by the National Natural Science
Foundation of China |
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| Abstract： |
Let $X_1,\cdots,X_N$ (where $N>m$) be independent
$N_m(\mu,\Si)$ random vectors, and put
$$\overline X=\frac 1{N}\dsize\sum_{i=1}^N X_i\ \roman{and}\ \ T'T=A=\dsize\sum_{i=1}^N (X_i-\overline X)(X_i-\overline X)',$$
where $T$ is upper- triangular with positive diagonal elements. The author
considers the problem of estimating $\Si$ ,and restricts his attention to
the class of estimates $\Bbb D=\{T'\bi^{\ast}T
+Nb^{\ast}\overline X\,\overline X$':$\bi^{\ast}$ is any diagonal matrix and
$ b^{\ast} $
is any nonnegative constant\} because it has the following attractive features:
(a) Its elements are all quadratic forms of the sufficient
and complete statistics $(\overline X,T)$.
(b) It contains all estimates of the form $aA+Nb\overline X\,\overline X'$
($a\ge 0$
and $b\ge 0$), which construct a complete subclass of the class of
nonnegative quadratic estimates $\Bbb D^{*}=\{X'BX: B\ge 0\}$
(where $X=(X_1,\cdots,X_N)'$) for any strict convex loss function.
(c) It contains all invariant estimates under the
transformation group of upper-triangular matrices.
The author obtains the characteristics for an estimate of the form
$$T'\bi T+ Nb\overline X\,\overline X'\,(\bi =\roman{diag}\{\de_1,\cdots,\de_m\}
\ge 0\ \ \roman{and}\ b\ge 0)$$
of $\Sigma $ to be admissible in $\Bbb D $ when the loss function is chosen
as tr$(\Si^{-1}\widehat \Si-I)^2$, and shows, by an example, that
$aA+Nb\overline X\,\overline X'$ ($a\ge 0$ and $b\ge 0$) is admissible in $\Bbb
D^* $ can not imply its admissibility in $\Bbb D$. |
Keywords： |
Covariance matrix, Admissible estimate, Bartlett's
decomposition. |
Classification： |
62C15, 62H12. |
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