Quantile/expectile regression, and extreme data analysis
Quantile/expectile regression, and extreme data analysis
Quantile/expectile regression, and extreme data analysis
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Expectile <strong>regression</strong><br />
Interrelationship between <strong>expectile</strong>s <strong>and</strong> quantiles† III<br />
Thus, G is the inverse of the <strong>expectile</strong> function, <strong>and</strong> its derivative is<br />
g(t) =<br />
µF (t) − P(t)<br />
{2(P(t) − tF (t)) + t − µ} 2 . (10)<br />
It can be shown that G is actually a distribution function (so that g is its<br />
density function). That is, the <strong>expectile</strong>s of F are precisely the quantiles<br />
of G defined here.<br />
Table: Density function, distribution function, <strong>and</strong> <strong>expectile</strong> function <strong>and</strong> r<strong>and</strong>om<br />
generation for the distribution associated with the <strong>expectile</strong>s of several<br />
st<strong>and</strong>ardized distributions. These functions are available in VGAM.<br />
Function<br />
[dpqr]eexp()<br />
[dpqr]ekoenker()<br />
[dpqr]enorm()<br />
[dpqr]eunif()<br />
Distribution<br />
Exponential<br />
Koenker<br />
Normal<br />
Uniform<br />
© T. W. Yee (University of Auckl<strong>and</strong>) <strong>Quantile</strong>/<strong>expectile</strong> <strong>regression</strong>, <strong>and</strong> <strong>extreme</strong> <strong>data</strong> <strong>analysis</strong> 18 July 2012 @ Cagliari 38/101/
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