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Abstract:
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Consider the scale estimation problem under the location-scale model X¹,X²,..,Xn~iid Fn((x-?)/s), where -8< ?< 8 and s>0 are unknown and Fn is an unknown member of the shrinking neighborhood Fen ={Fn│Fn=(1-en)Fo+enG, where en=k/Vn, k=cst>0, Fo known error distribution and G unknown} and the class of L-estimators of scale that are location-scale equivariant and Fisher consistent at Fo. On this class of estimators we derive the AMSE (maximal asymptotic mean-squared-error) in terms of the mixing distribution on the quantiles which defines the L-estimator. For non-Normal Fo, we next find estimators which have minimum AMSE over the subclass of (a) a - interquantile ranges and (b) mixtures of at most two a - interquantile ranges. We next derive the AMSE of the L-estimator of scale symmetrized about the median. When Fo symmetric, the symmetrized and non-symmetrized L-estimators have equal AMSE.
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