Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix, employing a matrix loss function

Heinz Neudecker (University of Amsterdam)

Consider $S\sim W_m (n, I_m, M'M)$. The habitual unbiased estimator of $M'M$ is $T:=S-nI_m$. Under certain conditions $T_{\alpha}:=T+\alpha({\rm tr}S)^{-1}I_m$ is better than $T$, for suitable $\alpha$. Leung (1994) showed this using the loss function

$$\lambda[(M'M)^{-1}, R]:={\rm tr}\{(M'M)^{-1}R-I_m\}^2.$$

We shall use a matrix loss function

$$L[(M'M)^{-1}, R]:=\{(M'M)^{-1}R-I_m\}'\{(M'M)^{-1}R-I_m\},$$

and apply Lywner partial ordering of symmetric matrices. An approximate domination result will be proved, the error term being of order $o(n^{-1})$. We shall use a matrix version of a Fundamental Identity for the noncentral Wishart distribution. [Leung gave a scalar version extending Hass's Fundamental Identity (scalar version) for the central Wishart distribution.] A matrix version of Leung's ancillary Lemma 3.1 will then be established. We shall employ an approximation of $\mathcal {E}({\rm tr} S)^{-1}S, \mathcal {E}$ being the expectation operator. A lemma of the matrix Hessian $\bigtriangledown\varphi F$, where $\varphi( F)$ is a scalar (matrix) function of $S$ will be proved. Further a lemma on the scalar Hessian tr ${\bigtriangledown F_2AF_1}$, where $F_1$ and $F_2$ are matrix functions of $S$ and $A$ is a constant matrix, will be given. References: Hass, L. R. (1981) Canadian J. Statist, 215-224. Leung, P. L. (1994) J. Multivariate Anal. 48, 107-14.