Baranchik-Type Estimators Under Modified Balanced Loss Functions

We study the problem of estimating the mean vector \(\theta = (\theta _{1}, \ldots , \theta _{d})\) of a random vector \(X \in \mathbb {R}^{d}\) that obeys a spherically symmetric distribution. We consider two types of modified balanced loss functions to overcome this problem: (1) The first type of loss function is defined as \(L_{\omega ,\delta _{0},\rho }(\delta ,\theta ) = \omega \rho (\|\delta - \delta _{0}\|^{2}) + (1-\omega ) \rho (\|\delta - \theta \|^{2})\) and (2) The second type of loss function is represented as \(\ell (\omega \|\delta - \delta _{0}\|^{2} + (1-\omega )\|\delta - \theta \|^{2})\) , where \(\delta _{0}\) is a target estimator of \(\theta \) and where \(\rho \) and \(\ell \) are increasing and concave functions. In the case when \(d \geq 4\) and when the target estimator \(\delta _{0}(X)=X\) , we establish a condition under which a class of Baranchik-type estimators \(\,\delta _{a, S}(X) = \left (1 - a(1-\omega ) S(\|X\|^2)/\|X\|^2\right )X\) dominates the maximum likelihood estimator \(\delta _{0}(X)\) and achieves minimaxity, where the function S is a twice differentiable almost everywhere function, and satisfying: \(\,\, 0 \leq S(.) \leq 1 , \,\,S(.)\neq 0, \,\,S'(.)\geq 0 \,\,\text{and} \,\, S''(.) \leq 0\) . These results extend the findings of Hobbad et al. to situations where the random vector X is not continuous with respect to the Lebesgue measure, meaning that it has no probability density function.