rbivcauchy {AoE}R Documentation

Random Number Generation for the Bivariate Cauchy Distribution

Description

Generates a random sample of the bivariate Cauchy distribution on the positive quadrant.

Usage

rbivcauchy(n)

Arguments

n Sample size.

Details

The density of the bivariate Cauchy distribution on the positive quadrant is given by

f(x, y) = (2/π) * (1 + x^2 + y^2)^(-3/2)

for x, y > 0. Its marginal distributions are the standard Cauchy distribution restricted to the positive half-line.

The bivariate Cauchy distribution is elliptic: a random pair (X, Y) with this distribution can be represented as

(X, Y) = (R cos Theta, R sin Theta)

where R > 0 and 0 <= Theta <= π/2 are independent random variables, P(R > r) = (1 + r^2)^(-1/2) for r > 0, and Theta is uniformly distributed on the interval [0, π/2].

The bivariate Cauchy distribution is in the bivariate max-domain of attraction of the bivariate extreme-value distribution with unit Frechet margins and with stable tail dependence function

l(x, y) = (x^2 + y^2)^(1/2)

for x, y > 0; see Einmahl et al. (2001). The angular or spectral measure with respect to the Euclidean norm is simply

Phi([0, theta]) = theta)

for 0 <= theta <= π/2.

Value

An n-by-2 matrix containing the generated data.

References

Einmahl, J.H.J., de Haan, L. and Piterbarg, V.I. (2001). Nonparametric estimation of the spectral measure of an extreme value distribution. The Annals of Statistics 29, 1401-1423.

Examples

x <- rbivcauchy(1000)
AngularMeasure(data = x, k = c(20, 30, 50))
abline(a = 0, b = 1, col = "red")

[Package AoE version 1.0.1 Index]