Shrinkage Estimation for Spherically Symmetric Distributions under Modified Balanced Loss Functions

Document Type : Original Article

Authors
Lahoucine Hobbad 1
Mohamed Alahiane 2
Idir Ouassou 1
Mustapha Rachdi 3
1 Cadi Ayyad University, National School of Applied Sciences, Av. Abdelkrim Khattabi, BP. 575, Marrakesh, Morocco
2 National School of Business and Management, Chouaib Doukkali University, Corner Avenue Ahmed Chaouki and Rue de Fés, B.P.122-24000, Eljadida, Morocco
3 Grenoble Alps University, Laboratory AGEIS EA 7407, Université Grenoble Alpes, UFR SHS, BP. 47, 38040 Grenoble Cedex09, France
10.22034/jirss.2025.2054911.1105
Abstract
We focus on the problem of estimating the average vector $\text{\boldmath$\text{\boldmath$\theta$}$}= ( \theta_{1} ,\ldots,\theta_{d} )$ of a random vector $\mathbf{X} \in \mathbb{R}^{d }$ which follows a spherically symmetric distribution. We consider modified balanced loss functions of the form:\\
$ \textbf{(i)}\ \ L_{\omega,\text{\boldmath$\text{\boldmath$\delta$}$}_0,\rho}(\text{\boldmath$\text{\boldmath$\delta$}$},\text{\boldmath$\text{\boldmath$\theta$}$})=\omega \rho(\|\text{\boldmath$\text{\boldmath$\delta$}$} - \text{\boldmath$\text{\boldmath$\delta$}$}_ {0 }\|^{2}) +(1-\omega)\rho(\|\text{\boldmath$\text{\boldmath$\delta$}$} - \text{\boldmath$\text{\boldmath$\theta$}$}\|^{2})\ \hbox{ and } \ \textbf{(ii)}\ \ \ell(\omega \ \|\text{\boldmath$\text{\boldmath$\delta$}$} - \text{\boldmath$\text{\boldmath$\delta$}$}_{0}\|^{2} +(1-\omega)\|\text{\boldmath$\text{\boldmath$\delta$}$}- \text{\boldmath$\text{\boldmath$\theta$}$}\|^{2}). $
Here, $\text{\boldmath$\text{\boldmath$\delta$}$}_{0}$ represents a target estimator of $\text{\boldmath$\text{\boldmath$\theta$}$}$, $\omega \in [0,1]$ and $\rho$ and $\ell$ are increasing and concave functions. If $d \geq 4$ and the target estimator is $\text{\boldmath$\text{\boldmath$\delta$}$}_{0}(\mathbf{X})=\mathbf{X}$, we provide conditions on the parameter $a$ for Baranchik type estimators $\text{\boldmath$\text{\boldmath$\delta$}$}_{a,S} (\mathbf{X}) =\left(1-a(1-\omega) S(\| \mathbf{X}\|^2)/\|\mathbf{\mathbf{X}}\|^{2} \right)\mathbf{X}$ and reach the minimaxity. These conditions are derived using the radial properties of spherically symmetric distributions, which do not require the existence of a probability density for the random vector $\mathbf{X}$. Furthermore, we extend the obtained results to the case of robust shrinkage estimators of the form $\text{\boldmath$\delta$}_{\omega,g}(\mathbf{X})=\mathbf{X} + a(1-\omega)g(\mathbf{X})$, where $g(\cdot)$ is a weakly differentiable and satisfying some conditions. Additionally, we conduct a simulation study in order to show the effectiveness and the usefulness of the obtained results.
Keywords
Modified Balanced Loss Function
Baranchik Type Estimators
Shrinkage Estimation
Dominance
Spherically Symmetric Distribution
Robust Schrinkage Estimators
Subjects
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Journal of the Iranian Statistical Society
Volume 23, Issue 2
December 2024
Pages 151-177

  • Receive Date 04 March 2025
  • Revise Date 12 May 2025
  • Accept Date 02 June 2025