Estimating quantiles with Linex loss function. Applications to VaR estimation

Volume 32 / 2005

Ryszard Zieliński Applicationes Mathematicae 32 (2005), 367-373 MSC: 62F10, 62G05. DOI: 10.4064/am32-4-1

Abstract

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.

Authors

  • Ryszard ZielińskiInstitute of Mathematics
    Polish Academy of Sciences
    P.O. Box 21
    00-956 Warszawa, Poland
    e-mail

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