Trimmed Mean and Huber's Estimator : Their Difference as a Goodness-of-Fit Criterion

J. Jureckova (1995). Trimmed Mean and Huber's Estimator : Their Difference as a Goodness-of-Fit Criterion. Journal of Statistical Research, Vol. 29, No. 2, pp. 31-35.

Abstract

For a sample $X_1\ldots,X_n$ from a symmetric (but unknown) distribution function $G(x-\theta)$ , we calculate the $\alpha$ -trimmed mean $L_{n\alpha}$ and the Huber estimator $M_{n\alpha}$ with $k=F^{-1}(\alpha)$ , $F$ being an hypothesis d.f. Then $L_{n\alpha}$ and $M_{n\alpha}$ are asymptotically equivalent provided $G^{-1}(1 - \alpha) = F^{-1}(1-\alpha) = -F^{-1}(\alpha) = - G^{-1}(\alpha)$ . We derive the asymptotic distribution of $n(L_{n\alpha}-M_{n\alpha})$ under this condition and more specifically, under the hypothesis $H^\star : G(x) = F(x)$ for $F^{-1}(\alpha)\leq x \leq F^{-1}(1-\alpha)$ . As an application, $n (L_{n\alpha}-M_{n\alpha})$ provides a test criterion for $H^\star$ .