Estimating quantiles with Linex loss function. Applications to VaR estimation

Ryszard Zieliński. "Estimating quantiles with Linex loss function. Applications to VaR estimation." Applicationes Mathematicae 32.4 (2005): 367-373. <http://eudml.org/doc/279061>.

@article{RyszardZieliński2005,
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 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.},
author = {Ryszard Zieliński},
journal = {Applicationes Mathematicae},
keywords = {quantile estimation; Linex loss; VaR (Value at Risk); normal distribution; nonparametric model},
language = {eng},
number = {4},
pages = {367-373},
title = {Estimating quantiles with Linex loss function. Applications to VaR estimation},
url = {http://eudml.org/doc/279061},
volume = {32},
year = {2005},
}

TY - JOUR
AU - Ryszard Zieliński
TI - Estimating quantiles with Linex loss function. Applications to VaR estimation
JO - Applicationes Mathematicae
PY - 2005
VL - 32
IS - 4
SP - 367
EP - 373
AB - 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.
LA - eng
KW - quantile estimation; Linex loss; VaR (Value at Risk); normal distribution; nonparametric model
UR - http://eudml.org/doc/279061
ER -