expectreg (version 0.25)

pkoenker: The "expectiles-meet-quantiles" distribution family.

Description

Density, distribution function, quantile function, random generation and expectile function for a family of distributions for which expectiles and quantiles coincide.

Usage

pemq(z,ncp=0,s=1)
demq(z,ncp=0,s=1)
qemq(q,ncp=0,s=1)
remq(n,ncp=0,s=1)
eemq(q,ncp=0,s=1)

Arguments

ncp

non centrality parameter and mean of the distribution.

s

scaling parameter, has to be positive.

z

vector of quantiles.

q

vector of asymmetries.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Value

• demq gives the density, pemq gives the distribution function, qemq gives the quantile function, eemq computes the expectiles numerically and is only provided for completeness, since the quantiles = expectiles can be determined analytically using qemq, and remq generates random deviates.

Details

This distribution has the cumulative distribution function: $F(x;ncp,s) = \frac{1}{2}(1 + sgn(\frac{x-ncp}{s}) \sqrt{1 - \frac{2}{2 + (\frac{x-ncp}{s})^2}})$ and the density: $f(x;ncp,s) = \frac{1}{s}( \frac{1}{2 + (\frac{x-ncp}{s})^2} )^\frac{3}{2}$ It has infinite variance, still can be scaled by the parameter s. It has mean ncp. In the canonical parameters it is equal to a students-t distribution with 2 degrees of freedom. For $s = \sqrt{2}$ it is equal to a distribution introduced by Koenker(2005).

References

Koenker R (2005) Quantile Regression Cambridge University Press, New York

See Also

enorm

Examples

x <- seq(-5,5,length=100)
plot(x,demq(x))
plot(x,pemq(x,ncp=1))

z <- remq(100,s=sqrt(2))

y <- seq(0,1,0.2)
qemq(y)
eemq(y)