Robust Estimation of a Location Parameter in the Presence of Asymmetry

Huber's theory of robust estimation of a location parameter is adapted to obtain estimators that are robust against a class of asymmetric departures from normality. Let F be a distribution function that is governed by the standard normal density on the set - d, d and is otherwise arbitrary. Let X₁,⋯, Xₙ be a random sample from F(x - θ), where θ is the unknown location parameter. If ψ is in a class of continuous skew-symmetric functions Ψc which vanish outside a certain set -c, c, then the estimator Tₙ, obtained by solving ∑ψ (Xᵢ - Tₙ) = 0 by Newton's method with the sample median as starting value, is a consistent estimator of θ. Also n1/2(Tₙ - θ) is asymptotically normal. For a model of symmetric contamination of the normal center of F, an asymptotic minimax variance problem is solved for the optimal ψ. The solution has the form ψ(x) = x for |x| x₀, ψ(x) = x₁ 1/2x₁(c - |x|) (x) for x₀ |x| c, and ψ(x) = 0 for |x| c. The results are extended to include an unknown scale parameter in the model.