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=(θ_1,…,θ_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 (θ,0). The loss functions are assumed to be modified balanced loss functions of the form: (i) ωρ(|(|δ-δ_0 | )|^2 )+(1-ω)ρ(|(|δ-θ| )|^2 ) and (ii) l(ω|(|δ-δ_0 | )|^2+(1-ω)〖||δ-θ||〗^2) where δ_0 is a target estimation of θ, and where ρ and l are increasing and concave functions. In the case when d≥4 and the target estimator δ_0 (X)=X, we establish a conditions under which the estimators of the form δ_(ω,g) (X,|(|U| )|^2 )=X+a|(|U| )|^2 (1-ω)g(X) dominate δ_0 (X)=X and are minimax, where we suppose there exists a nonpositive function h(.) such that h(X)is subharmonic and E_(R,θ) [R^2 h(ω)] is noincreasing with W~U_(R,θ),E_θ [|h(X) | ]<∞and such that 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