On the Dependence between Functions of Quantile and Dispersion Estimators

Abstract: In this paper, we derive the joint asymptotic distributions of functions of quantile estimators (the non-parametric sample quantile and the parametric location-scale quantile estimator) with functions of measure of dispersion estimators (the sample variance, sample mean absolute deviation, sample median absolute deviation) - assuming an underlying identically and independently distributed sample. We also discuss the conditions required by the use of such estimators. Further, we show that these results can be extended to any higher order absolute central sample moment as measure of dispersion. Aware of the difference in speed of convergence of the two quantile estimators, we compare the impact of the choice of the quantile estimator (and measure of dispersion) on the asymptotic correlations. Then we prove a scaling law for the asymptotic dependence of quantile estimators with measure of dispersion estimators. Finally, we show a good finite sample performance of the asymptotics in simulations for elliptical distributions. All the results should constitute an important and useful complement in the statistical literature as those estimators are either of standard use in statistics and application fields, or should become as such because of weaker conditions in the asymptotic theorems.