Rate of convergence of intermediate order statistics (Q677878)

Let \(X_1,\dots, X_n\) be an i.i.d. sample and \(X_{1:n}\leq\dots\leq X_{n:n}\) be its order statistic. If \(k=k_n\to\infty\) and \(k/n\to 0\), then \(X_{n-k+1:n}\) is called an intermediate order statistic. Under various conditions, the exact convergence rates are derived for the uniform convergence of the density of the intermediate statistics towards a normal or log-normal density. Some convergence rates in uniform metric and in the total variation metric are also given.