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.