We consider the problem of estimating a d-dimensional parameter \(\theta =(\theta _1, \ldots , \theta _d)\) when the observation is a \(d+k\) vector \((X, U)\) where \(\dim X = d\) and where U is a residual vector with \(\dim U=k\) . The distributional assumption is that \((X,U)\) has a spherically symmetric distribution around \((\theta ,0)\) . The loss functions are assumed to be modified balanced loss functions of the following forms: (i) \(\omega \rho (\|\delta - \delta _{0}\|^{2}) +(1-\omega )\rho (\|\delta - \theta \|^{2})\) and (ii) \(\ell (\omega \|\delta - \delta _{0}\|^{2} +(1-\omega )\|\delta - \theta \|^{2})\) , where \(\delta _{0}\) is a target estimation of \(\theta \) and \(\rho \) and \(\ell \) are increasing and concave functions. In the case when \(d \geq 4\) and the target estimator \(\delta _{0}(X) = X\) , we establish conditions under which the estimators of the form \(\delta _{\omega ,g}(X,\|U\|^{2}) = X + a\|U\|^{2}(1-\omega )g(X)\) dominate \(\delta _{0}(X)=X\) and are minimax, where we suppose there exists a nonpositive function \(h(\cdot )\) such that \(h(X)\) is subharmonic and \(E_{R,\theta }\left [ R^{2}h(W)\right ]\) is nonincreasing with \(W\sim U_{R,\theta }\) , \(\operatorname {E}_{\theta }\left [ |h(X)|\right ] < \infty \) and such that \(g(X)\) is weakly differentiable and also satisfies (a) \(\text{div}(g(X)) \leq h(X)\) and (b) \(\|g(X)\|^{2} + 2h(X) \leq 0\) .