Approximation of intermediate quantile processes
Let \(X_ 1,X_ 2,...,X_ n\) be independent real random variables with common distribution F, and let \(X_{1,n}\leq X_{2,n}\leq...\leq X_{n,n}\) denote their order statistics. Let \(\{k_ n:\) \(n\geq 1\}\) be a sequence of positive integers such that \(k_ n\leq n\), and \(k_ n/n\to 0\) as \(n\to \infty\). This paper proves the weak convergence in weighted metrics of the intermediate quantile process \[ \gamma_ n(t)=nk_ n^{-1/2}(k_ n/n)^{\alpha} L(k_ n/n)t^{\alpha}(Q(tk_ n/n)-Q_ n(tk_ n/n)) \] to a Wiener process. Here \(\lambda /k_ n\leq t\leq T\), L is a slowly varying function at 0, and Q \((Q_ n)\) is the (empirical) quantile function. F is required to be continuous on its open support \((t_ F,t^ F)\), and to have a strictly positive density, f, on \((t_ F,x_ 0]\) (for a \(x_ 0>t_ F)\). Also, \(f(Q(t))=t^{\alpha}L(t).\) The proof combines a strong approximation of quantile processes due to the authors, \textit{S. Csörgö}, and \textit{D. M. Mason} [Ann. Probab. 14, 31-85 (1986; Zbl 0589.60029)], the analysis of slowly varying functions, and several probability estimates for intermediate quantile processes.
strong approximation of quantile processes
scientific article; zbMATH DE number 3994648
scientific article