Estimating quantiles with Linex loss function. Applications to VaR estimation

Ryszard Zieliński¹

¹ Institute of Mathematics Polish Academy of Sciences P.O. Box 21 00-956 Warszawa, Poland

Applicationes Mathematicae, Tome 32 (2005) no. 4, pp. 367-373

Résumé

Sometimes, e.g. in the context of estimating VaR (Value at Risk), underestimating a quantile is less desirable than overestimating it, which suggests measuring the error of estimation by an asymmetric loss function. As a loss function when estimating a parameter $\theta$ by an estimator $T$ we take the well known Linex function $\exp \{\alpha(T-\theta)\}-\alpha(T-\theta)-1$. To estimate the quantile of order $q\in(0,1)$ of a normal distribution $N(\mu,\sigma)$, we construct an optimal estimator in the class of all estimators of the form $\overline x+k\sigma$, $-\infty k \infty$, if $\sigma$ is known, or of the form $\overline x + \lambda s$, if both parameters $\mu$ and $\sigma$ are unknown; here $\overline x$ and $s$ are the standard estimators of $\mu$ and $\sigma$, respectively. To estimate a quantile of an unknown distribution $F$ from the family $\cal F$ of all continuous and strictly increasing distribution functions we construct an optimal estimator in the class $\cal T$ of all estimators which are equivariant with respect to monotone transformations of data.

Détail

DOI : 10.4064/am32-4-1

Keywords: sometimes context estimating var value risk underestimating quantile desirable overestimating which suggests measuring error estimation asymmetric loss function loss function estimating parameter theta estimator known linex function exp alpha t theta alpha t theta estimate quantile order normal distribution sigma construct optimal estimator class estimators form overline sigma infty infty sigma known form overline lambda parameters sigma unknown here overline standard estimators sigma respectively estimate quantile unknown distribution family cal continuous strictly increasing distribution functions construct optimal estimator class cal estimators which equivariant respect monotone transformations