The Length of the Shorth
The Length of the Shorth
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 …