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Sometimes, e.g. in the context of estimating V aR (Value at Risk), underestimating a quantile is less desirable than overestimating it which suggest to measure the error of estimation by an asymmetric loss function. As a loss function when estimat- ing a parameter í by an estimator T we take the well known Linex Function expfã(T Ä í)g Ä ã(T Ä í) Ä 1. To estimate the quan- tile of order q 2 (0;1) of a normal distribution N(ñ;õ), we con- struct the optimal estimator in the class of all estimators of the form ¯ x + kõ; Ä1 < k < 1, if õ is known, or of the formx + ïs, if both parameters ñ and õ are unknown; here ¯ x and s are standard estima- tors of ñ and õ, respectively. To estimate a quantile of an unknown distribution F from the family F of all continuous and strictly in- creasing distribution functions we construct the optimal estimator in the class T of all estimators which are equivariant with respect to monotone transformations of data.

Estimating quantiles with Linex loss function. Applications to VaR estimation