Estimation under ℓ1-Symmetry

The estimation of the location parameter of an ℓ₁-symmetric distribution is considered. Specifically when a p-dimensional random vector has a distribution that is a mixture of uniform distributions on the ℓ₁-sphere, we investigate a general class of estimators of the form δ=X+g. Under the usual quadratic loss, domination of δ over X is obtained through the partial differential inequality 4 div g+2X ∂²g+ g²0 and a new superharmonicity-type-like notion adapted to the ℓ₁-context. Specifically the condition of ℓ₁-superharmonicity is that 2Δf+X ³f0 and div ³f0 as compared to the usual (ℓ₂) condition Δf0.