# enorm: Expectiles of distributions In expectreg: Expectile and Quantile Regression

## Description

Much like the 0.5 quantile of a distribution is the median, the 0.5 expectile is the mean / expected value. These functions add the possibility of calculating expectiles of known distributions. The functions starting with 'e' calculate an expectile value for given asymmetry values, the functions starting with 'pe' calculate vice versa.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 enorm(asy, m = 0, sd = 1) penorm(e, m = 0, sd = 1) ebeta(asy, a = 1, b = 1) pebeta(e, a = 1, b = 1) eunif(asy, min = 0, max = 1) peunif(e, min = 0, max = 1) et(asy, df) pet(e, df) elnorm(asy, meanlog = 0, sdlog = 1) pelnorm(e, meanlog = 0, sdlog = 1) egamma(asy, shape, rate = 1, scale = 1/rate) pegamma(e, shape, rate = 1, scale = 1/rate) eexp(asy, rate = 1) peexp(e, rate = 1) echisq(asy, df) pechisq(e, df) 

## Arguments

 asy vector of asymmetries with values between 0 and 1. e vector of expectiles from the respective distribution. m, sd mean and standard deviation of the Normal distribution. a, b positive parameters of the Beta distribution. min, max minimum, maximum of the uniform distribution. df degrees of freedom of the student t and chi squared distribution. meanlog, sdlog parameters of the lognormal distribution. shape, rate, scale parameters of the gamma distribution (with 2 different parametrizations) and parameter of the exponential distribution which is a special case of the gamma with shape=1.

## Details

An expectile of a distribution cannot be determined explicitely, but instead is given by an equation. The expectile z for an asymmetry p is: p = \frac{G(z) - z F(z)}{2(G(z) - z F(z)) + z - m} where m is the mean, F the cdf and G the partial moment function G(z) = \int\limits_{-∞}^{z} uf(u) \mbox{d}u .

## Value

Vector of the expectiles or asymmetry values for the desired distribution.

## Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
http://www.uni-Oldenburg.de

Thomas Kneib
Georg August University Goettingen
http://www.uni-goettingen.de

## References

Newey W and Powell J (1987) Asymmetric least squares estimation and testing Econometrica, 55:819-847

eemq
 1 2 3 4 5 x <- seq(0.02,0.98,0.2) e = enorm(x) e penorm(e)