Improving location estimators for spherically symmetric models with residual under modified balanced loss functions

Alahiane, Mohamed; Hobbad, Lahoucine; Arjdal, Ismail; Ouassou, Idir

doi:10.22034/jirss.2026.2077431.1162

Improving location estimators for spherically symmetric models with residual under modified balanced loss functions

Articles in Press

Document Type : Original Article

Authors

Mohamed Alahiane ¹

Lahoucine Hobbad ²

Ismail Arjdal ³

Idir Ouassou ²

¹ National School of Business and Management, Chouaib Doukkali University, Corner Avenue Ahmed Chaouki and Rue de Fés, B.P.122-24000, Eljadida, Morocco

² Cadi Ayyad University, National School of Applied Sciences, Av. Abdelkrim Khattabi, BP. 575, Marrakesh, Morocco.

³ Chouaib Doukkali University, Faculty of Sciences, Department of Mathematics, 24000, El-Jadida, Morocco.

10.22034/jirss.2026.2077431.1162

Abstract

We consider the problem of estimating a d-dimensional parameter θ=(θ₁,…,θ_d ) when the observation is a d + k dimensional vector (X,U) where dim⁡X=d and U is a residual vector with dim ⁡U=k. The distributional assumption is that (X,U) has a spherically symmetric distribution around (θ,0). The loss functions are assumed to be modified balanced loss functions of the form: (i) ωρ(∥δ-δ₀ ∥² )+(1-ω)ρ(∥δ-θ∥² ) and (ii) l(ω∥δ-δ₀ ∥²+(1-ω)∥δ-θ∥²) where δ₀ is a target estimator for θ, and where ρ and l are increasing and concave functions. In the case when d≥4 and the target estimator δ₀(X)=X, we establish conditions under which the estimators of the form δ_ω,g(X,∥U∥² )=X+a∥U∥² (1-ω)g(X) dominate δ₀ (X)=X and are minimax, where we suppose that there exists a nonpositive function h(.) such that h(X) is subharmonic, E_R,θ[R² h(W)] is nonincreasing with W∼U_R,θ , E_θ [∣h(X)∣]<∞ and the function g(X) is weakly differentiable and also satisfies (a) div⁡(g(X))≤h(X), (b) ∥g(X) ∥²+2h(X)≤0.

Keywords

Modified balanced loss

Baranchik type estimators

Dominance

James-Stein estimator

Dirichlet distribution

spherically symmetric distribution with residual

Subjects

Statistical Modelling

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