rotations (version 1.6.5)

Cayley: The symmetric Cayley distribution

Description

Density, distribution function and random generation for the Cayley distribution with concentration kappa \(\kappa\).

Usage

dcayley(r, kappa = 1, nu = NULL, Haar = TRUE)
pcayley(q, kappa = 1, nu = NULL, lower.tail = TRUE)
rcayley(n, kappa = 1, nu = NULL)

Value

dcayley

gives the density

pcayley

gives the distribution function

rcayley

generates a vector of random deviates

Arguments

r, q

vector of quantiles.

kappa

concentration parameter.

nu

circular variance, can be used in place of kappa.

Haar

logical; if TRUE density is evaluated with respect to the Haar measure.

lower.tail

logical; if TRUE (default) probabilities are \(P(X\leq x)\) otherwise, \(P(X>x)\).

n

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

Details

The symmetric Cayley distribution with concentration \(\kappa\) has density $$C_C(r |\kappa)=\frac{1}{\sqrt{\pi}} \frac{\Gamma(\kappa+2)}{\Gamma(\kappa+1/2)}2^{-(\kappa+1)}(1+\cos r)^\kappa(1-\cos r).$$ The Cayley distribution is equivalent to the de la Vallee Poussin distribution of Schaeben1997.

Schaeben1997 leon2006

See Also

Angular-distributions for other distributions in the rotations package.

Examples

r <- seq(-pi, pi, length = 500)

#Visualize the Cayley density fucntion with respect to the Haar measure
plot(r, dcayley(r, kappa = 10), type = "l", ylab = "f(r)")

#Visualize the Cayley density fucntion with respect to the Lebesgue measure
plot(r, dcayley(r, kappa = 10, Haar = FALSE), type = "l", ylab = "f(r)")

#Plot the Cayley CDF
plot(r,pcayley(r,kappa = 10), type = "l", ylab = "F(r)")

#Generate random observations from Cayley distribution
rs <- rcayley(20, kappa = 1)
hist(rs, breaks = 10)