Shrinkage Estimation for Spherically Symmetric Distributions under Modified Balanced Loss Functions

Abstract

We focus on the problem of estimating the average vector θ = (θ 1 , . . . , θ d ) of a random vector X ∈ R d which follows a spherically symmetric distribution. We consider modified balanced loss functions of the form:

Here, δ 0 represents a target estimator of θ, ω ∈ [0, 1] and ρ and ℓ are increasing and concave functions. If d ≥ 4 and the target estimator is δ 0 (X) = X, we provide conditions on the parameter a for Baranchik type estimators δ a,S (X) = 1a(1 -ω)S( X 2 )/ X 2 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 X. Furthermore, we extend the obtained results to the case of robust shrinkage estimators of the form δ ω,g (X) = X + a(1 -ω)g(X), where g(•) 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.

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BibTeX RIS

BibTeX

@article{hobbad2026shrinkage,
  title = {{Shrinkage Estimation for Spherically Symmetric Distributions under Modified Balanced Loss Functions}},
  author = {Lahoucine Hobbad and Mohamed Alahiane and Idir Ouassou and Mustapha Mr Rachdi},
  journal = {Journal of The Iranian Statistical Society},
  year = {2026},
  month = sep,
  doi = {10.22034/jirss.2025.2054911.1105},
  url = {https://hal.science/hal-05756861v1},
  note = {CC BY-NC 4.0},
}

RIS

TY  - JOUR
TI  - Shrinkage Estimation for Spherically Symmetric Distributions under Modified Balanced Loss Functions
AU  - Lahoucine Hobbad
AU  - Mohamed Alahiane
AU  - Idir Ouassou
AU  - Mustapha Mr Rachdi
PY  - 2026
DA  - 2026/09/21
JO  - Journal of The Iranian Statistical Society
DO  - 10.22034/jirss.2025.2054911.1105
UR  - https://hal.science/hal-05756861v1
AB  - We focus on the problem of estimating the average vector θ = (θ 1 , . . . , θ d ) of a random vector X ∈ R d which follows a spherically symmetric distribution. We consider modified balanced loss functions of the form:  Here, δ 0 represents a target estimator of θ, ω ∈ [0, 1] and ρ and ℓ are increasing and concave functions. If d ≥ 4 and the target estimator is δ 0 (X) = X, we provide conditions on the parameter a for Baranchik type estimators δ a,S (X) = 1a(1 -ω)S( X 2 )/ X 2 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 X. Furthermore, we extend the obtained results to the case of robust shrinkage estimators of the form δ ω,g (X) = X + a(1 -ω)g(X), where g(•) 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.
SN  - 1726-4057
SN  - 2538-189X
ER  -

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