RTDE (version 0.2-2)

Frank: The Frank Distribution

Description

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

Usage

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)

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.

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 \(P[X \le x]\), otherwise, \(P[X > x]\).

Author

Christophe Dutang

Details

The Frank is defined by the following distribution function $$ C(u,v) = - \frac{1}{\alpha} \log\left[1-\frac{(1-e^{-\alpha u})(1-e^{-\alpha v}) }{ 1-e^{-\alpha}}\right], $$ for all \(u,v\) in [0,1]. When lower.tail=FALSE, pfrank returns the survival copula \(P(U > u, V > v)\).

References

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

Examples

#####
# (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))