Shrinkage Estimation for Spherically Symmetric Distributions under Modified Balanced Loss Functions
Year: 2024
Type: article
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.
Source: HAL
Institutions: Cadi Ayyad University, Université Grenoble Alpes
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214395161