pemq | R Documentation |

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

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(asy,ncp=0,s=1) peemq(e,ncp=0,s=1)

`ncp` |
non centrality parameter and mean of the distribution. |

`s` |
scaling parameter, has to be positive. |

`z, e` |
vector of quantiles / expectiles. |

`q, asy` |
vector of asymmetries / probabilities. |

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

This distribution has the cumulative distribution function:
* F(x;ncp,s) = \frac{1}{2}(1 + sgn(\frac{x-ncp}{s}) √{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 = √{2} * it is equal to a distribution introduced by Koenker(2005).

`demq`

gives the density, `pemq`

and `peemq`

give 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.

Fabian Otto- Sobotka

Carl von Ossietzky University Oldenburg

https://uol.de

Thomas Kneib

Georg August University Goettingen

https://www.uni-goettingen.de

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

`enorm`

x <- seq(-5,5,length=100) plot(x,demq(x)) plot(x,pemq(x,ncp=1)) z <- remq(100,s=sqrt(2)) plot(z) y <- seq(0.02,0.98,0.2) qemq(y) eemq(y) pemq(x) - peemq(x)

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