The Length of the Shorth

Let Ĥₙ(α) (0 < α < 1) denote the length of the shortest α-fraction of the ordered sample X1:n, X2:n, ⋯, Xn:n, i.e., Ĥₙ(α) = min + j:n - Xk:n: 1 ≤ k ≤ k + j ≤ n; (j + 1)/n ≥ α\. Such quantities arise in the context of robust scale estimation. Using the concept of compact derivatives of statistical functionals, the asymptotic behaviour of Ĥₙ(α) as n → ∞ is investigated.