On the Dependence between Quantiles and Dispersion Estimators

Bräutigam Marcel and Kratz Marie ()
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Bräutigam Marcel: Sorbonne University
Kratz Marie: ESSEC Research Center, ESSEC Business School, Postal: ESSEC Research Center, BP 105, 95021 Cergy, France

No WP1807, ESSEC Working Papers from ESSEC Research Center, ESSEC Business School

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)
JEL-codes: C13 C14 C30 C58 C69 G32 (search for similar items in EconPapers)
Pages: 90 pages
Date: 2018-12
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Persistent link: https://EconPapers.repec.org/RePEc:ebg:essewp:dr-18007

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