On the Dependence between Quantiles and Dispersion Estimators

Marcel Bräutigam and Marie Kratz ()
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Marcel Bräutigam: LabEx MME-DII - UCP - Université de Cergy Pontoise - Université Paris-Seine, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique, CREAR - Center of Research in Econo-finance and Actuarial sciences on Risk / Centre de Recherche Econo-financière et Actuarielle sur le Risque - ESSEC Business School
Marie Kratz: LabEx MME-DII - UCP - Université de Cergy Pontoise - Université Paris-Seine, CREAR - Center of Research in Econo-finance and Actuarial sciences on Risk / Centre de Recherche Econo-financière et Actuarielle sur le Risque - ESSEC Business School, ESSEC Business School

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Abstract: In this study, we derive the joint asymptotic distributions of functionals of quantile estimators (the non-parametric sample quantile and the parametric location-scale quantile) and functionals of measure of dispersion estimators (the sample standard deviation, sample mean absolute deviation, sample median absolute deviation) - assuming an underlying identically and independently distributed sample. Additionally, for location-scale distributions, we show that asymptotic correlations of such functionals do not depend on the mean and variance parameter of the distribution. Further, we compare the impact of the choice of the quantile estimator (sample quantile vs. parametric location-scale quantile) in terms of speed of convergence of the asymptotic covariance and correlations respectively. As application, we show in simulations a good finite sample performance of the asymptotics. Further, we show how the theoretical dependence results can be applied to the most well-known risk measures (Value-at-Risk, Expected Shortfall, expectile). Finally, we relate the theoretical results to empirical findings in the literature of the dependence between risk measure prediction (on historical samples) and the estimated volatility.

Keywords: asymptotic distribution; sample quantile; measure of dispersion; non-linear dependence; VaR; ES; correlation (search for similar items in EconPapers)
Date: 2018-12
New Economics Papers: this item is included in nep-ecm and nep-rmg
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