Asymptotic joint distribution of sample quantiles and sample mean with applications

Based on a sample from an absolutely continuous distribution F with density f, and with the aid of the Bahadur (Ann. Math. Statist. 37( 1966 ), 577-580) representation of sample quantiles, the asymptotic joint distribution of three statistics, the sample p^th and q^th quantiles (0 < p < q < l) and the sample mean, is obtained. Using the Cramer-Wold device, asymptotic distributions of functions of the three statistics can be derived. In particular, the asymptotic joint distribution of the ratio of sample p^th quantile to sample mean and the ratio of sample q^th quantile to sample mean is presented. Finally, consistent estimators are proposed for the variances and covariances of these limiting distributions.