# Frank: The Frank Distribution In RTDE: Robust Tail Dependence Estimation

## Description

Density function, distribution function, quantile function, random generation.

## Usage

 ```1 2 3 4``` ```dfrank(u, v, alpha, log = FALSE) pfrank(u, v, alpha, lower.tail=TRUE, log.p = FALSE) qfrank(p, alpha, lower.tail=TRUE, log.p = FALSE) rfrank(n, alpha) ```

## Arguments

 `u, v` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `alpha` shape parameter. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

## Details

The Frank is defined by the following distribution function

C(u,v) = - 1/α *log( 1-(1-exp(-α u))*(1-exp(-α v)) /( 1-exp(-α)) ),

for all u,v in [0,1]. When `lower.tail=FALSE`, `pfrank` returns the survival copula P(U > u, V > v).

## Value

`dfrank` gives the density, `pfrank` gives the distribution function, `qfrank` gives the quantile function, and `rfrank` generates random deviates.

The length of the result is determined by `n` for `rfrank`, and is the maximum of the lengths of the numerical parameters for the other functions.

The numerical parameters other than `n` are recycled to the length of the result. Only the first elements of the logical parameters are used.

## Author(s)

Christophe Dutang

## References

Nelsen, R. (2006), An Introduction to Copula, Second Edition, Springer.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```##### # (1) density function u <- v <- seq(0, 1, length=25) cbind(u, v, dfrank(u, v, 1/2)) cbind(u, v, outer(u, v, dfrank, alpha=1/2)) ##### # (2) distribution function cbind(u, v, pfrank(u, v, 1/2)) cbind(u, v, outer(u, v, pfrank, alpha=1/2)) ```

RTDE documentation built on Jan. 8, 2020, 5:09 p.m.