Some Observations on Robust Estimation

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Abstract
Let Tj be a reasonable estimator (for example, a minimum mean square error estimator) of the parameter θ of the family Dj of distributions, j = 1, 2, …, m. An estimator T, which is a weighted mean of T 1, T s, …, Tm , is found that has the same asymptotic distribution as that of Tj , when the sample comes from Dj , j = 1, 2, …, m. Here the weights are functions of the sample items. Empirical evidence is given which indicates that T is satisfactory for small sample sizes. It is proved that if Tj and the weight Wj are odd location and even location-free statistics, respectively, j = 1, 2, …, m, then T = ΣWiTi , where ΣWi = 1, is an unbiased estimator of the center of every symmetric distribution, provided certain expectations exist. This is useful in the construction of the weight function Wj.

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