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+2Xċ∂2g+ ‖g‖2⩽0 and a new superharmonicity-type-like notion adapted to the ℓ1-context. Specifically the condition of ℓ1-superharmonicity is that 2Δf+Xċ∇3f⩽0 and div ∇3f⩾0 as compared to the usual (ℓ2) condition Δf⩽0.