evd (version 1.2-3)

bvhr: The Husler-Reiss Model

Description

Density, distribution function and random generation for the Husler-Reiss model.

Usage

dbvhr(x, dep, mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) 
pbvhr(q, dep, mar1 = c(0, 1, 0), mar2 = mar1) 
rbvhr(n, dep, mar1 = c(0, 1, 0), mar2 = mar1)

Arguments

x, q

A vector of length two or a matrix with two columns, in which case the density/distribution is evaluated across the rows.

n

Number of observations.

dep

Dependence parameter.

mar1, mar2

Vectors of length three containing marginal parameters, or matrices with three columns where each column represents a vector of values to be passed to the corresponding marginal parameter.

log

Logical; if TRUE, the log density is returned.

Value

• dbvhr gives the density, pbvhr gives the distribution function and rbvhr generates random deviates.

Details

The Husler-Reiss distribution function (Husler and Reiss, 1989) with parameter $\code{dep} = r$ is $$G(z_1,z_2) = \exp\left(-y_1\Phi{r^{-1}+{\textstyle\frac{1}{2}} r[\log(y_1/y_2)]} - y_2\Phi{r^{-1}+{\textstyle\frac{1}{2}}r [\log(y_2/y_1)]}\right)$$ where $\Phi(\cdot)$ is the standard normal distribution function, $r > 0$ and $$y_i = {1+s_i(z_i-a_i)/b_i}^{-1/s_i}$$ for $1+s_i(z_i-a_i)/b_i > 0$ and $i = 1,2$, where the marginal parameters are given by $\code{mari} = (a_i,b_i,s_i)$, $b_i > 0$. If $s_i = 0$ then $y_i$ is defined by continuity. The univariate marginal distributions are generalized extreme value.

Independence is obtained in the limit as $r$ approaches zero. Complete dependence is obtained as $r$ tends to infinity.

References

Husler, J. and Reiss, R.-D. (1989) Maxima of normal random vectors: between independence and complete dependence. Statist. Probab. Letters, 7, 283--286.

See Also

abvhr, rgev

Examples

dbvhr(matrix(rep(0:4,2),ncol=2), 1.7)
pbvhr(matrix(rep(0:4,2),ncol=2), 1.7)  
rbvhr(10, 1.7)