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 θ by an estimator T we take the well known Linex function exp{α(T-θ)} - α(T-θ) - 1. To estimate the quantile of order q ∈ (0,1) of a normal distribution N(μ,σ), we construct an optimal estimator in the class of all estimators of the form x̅ + kσ, -∞ < k < ∞, if σ is known, or of the form x̅ + λs, if both parameters μ and σ are unknown; here x̅ and s are the standard estimators of μ and σ, respectively. To estimate a quantile of an unknown distribution F from the family ℱ of all continuous and strictly increasing distribution functions we construct an optimal estimator in the class 𝓣 of all estimators which are equivariant with respect to monotone transformations of data.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am32-4-1,
author = {Ryszard Zieli\'nski},
title = {Estimating quantiles with Linex loss function. Applications to VaR estimation},
journal = {Applicationes Mathematicae},
volume = {32},
year = {2005},
pages = {367-373},
zbl = {1086.62030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am32-4-1}
}
Ryszard Zieliński. Estimating quantiles with Linex loss function. Applications to VaR estimation. Applicationes Mathematicae, Tome 32 (2005) pp. 367-373. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am32-4-1/