rbivcauchy {AoE} | R Documentation |
Generates a random sample of the bivariate Cauchy distribution on the positive quadrant.
rbivcauchy(n)
n |
Sample size. |
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.
An n-by-2 matrix containing the generated data.
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.
x <- rbivcauchy(1000) AngularMeasure(data = x, k = c(20, 30, 50)) abline(a = 0, b = 1, col = "red")