Estimating the quantile function of a location-scale family of distributions based on few selected order statistics
Let \(X_ 1,...,X_ n\) be i.i.d. random variables with d.f. F where \(F(x)=F_ 0((x-\mu)/\sigma)\) with \(F_ 0\) being a known d.f. and \(\mu\) and \(\sigma\) being unknown location and scale parameters. Given a positive integer k and numbers \(0<p_ 1<p_ 2<...<p_ k<1\) denote by \(X_{(n_ 1)}\leq...\leq X_{(n_ k)}\) the order statistics of \(X_ 1,...,X_ n\) with indices \(n_ i=[np_ i]+1\). By making use of the asymptotic joint normality of \(X_{(n_ 1)},...,X_{(n_ k)}\) one can establish an asymptotically best linear unbiased estimator (ABLUE) of the underlying \(\xi\)-quantile for every fixed vector \((p_ 1,...,p_ k).\) The main purpose of the present article is to find those \(p_ 1,...,p_ k\) which minimize the asymptotic variance over the class of ABLUE's in the particular case of exponential r.v.'s. Further results are obtained for Pareto and double exponential r.v.'s.
asymptotically best linear unbiased estimator
scientific article; zbMATH DE number 3846671