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 n12(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)=x1tanh[12x1(c−|x|)]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.
Citation
John R. Collins. "Robust Estimation of a Location Parameter in the Presence of Asymmetry." Ann. Statist. 4 (1) 68 - 85, January, 1976. https://doi.org/10.1214/aos/1176343348
Information