The Length of the Shorth

Abstract

Let $\hat{H}_n(\alpha) (0 < \alpha < 1)$ denote the length of the shortest $\alpha$-fraction of the ordered sample $X_{1:n}, X_{2:n}, \cdots, X_{n:n}$, i.e., $\hat{H}_n(\alpha) = \min\{X_{k + j:n} - X_{k:n}: 1 \leq k \leq k + j \leq n; (j + 1)/n \geq \alpha\}.$ Such quantities arise in the context of robust scale estimation. Using the concept of compact derivatives of statistical functionals, the asymptotic behaviour of $\hat{H}_n(\alpha)$ as $n \rightarrow \infty$ is investigated.