Robust Estimation of a Location Parameter in the Presence of Asymmetry

Abstract

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 $|x| \leqq x_0, \psi(x) = x_1 \tanh \lbrack\frac{1}{2}x_1(c - |x|)\rbrack\operatorname{sgn} (x)$ for x0 ≤ |x| ≤ c, and ψ(x) = 0 for |x| ≥ c. The results are extended to include an unknown scale parameter in the model.