A note on double k-class estimators under elliptical symmetry

In this paper, estimation of the regression vector parameter in the multiple regression
model $y = \bm{X}\beta +\epsilon$ is considered, when the error term belongs to the
class of elliptically contoured distributions (ECD), say, $\epsilon\sim EC_n(0, \sigma^2\bm{V}, \psi)$ ,
where $\sigma^2$ is unknown and $\bm{V}$ is a symmetric p.d known matrix with the characteristic
generator $\psi$ . It is well-known that UMVU estimator of $\beta$ has the
form $(\bm{X}^'\bm{V}^{-}\bm{X})^{-1}\bm{X} \bm{V}^{-1}y$ . In this paper using integral series representation of ECDs, the dominance conditions of double k-class estimators given by

$\begin{align*} \hat{\beta}_{ k_1,k_2} = \Big[1-\frac{k_1\hat{\epsilon}^' \bm{V}^{-1}\hat{\epsilon}}{y^'y-k_2\hat{\epsilon}^'\bm{V}^{-1}\hat{\epsilon}} \end{align*}$

over UMVUE, have been derived under weighted quadratic loss function.

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