evd (version 1.2-3)

bvaneglog: The Bivariate Asymmetric Negative Logistic Model

Description

Density, distribution function and random generation for the bivariate asymmetric negative logistic model.

Usage

dbvaneglog(x, dep, asy = c(1, 1), mar1 = c(0, 1, 0), mar2 = mar1,
    log = FALSE) 
pbvaneglog(q, dep, asy = c(1, 1), mar1 = c(0, 1, 0), mar2 = mar1) 
rbvaneglog(n, dep, asy = c(1, 1), 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.

asy

A vector containing the two asymmetry parameters.

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

• dbvaneglog gives the density, pbvaneglog gives the distribution function and rbvaneglog generates random deviates.

Details

The bivariate asymmetric negative logistic distribution function with parameters parameters $\code{dep} = r$ and $\code{asy} = (t_1,t_2)$ is $$G(z_1,z_2) = \exp\left{-y_1-y_2+ [(t_1y_1)^{-r}+(t_2y_2)^{-r}]^{-1/r}\right}$$ where $r > 0$, $0 < t_1,t_2 \leq 1$, 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.

When $t_1 = t_2 = 1$ the asymmetric negative logistic model is equivalent to the negative logistic model. Independence is obtained in the limit as either $r$, $t_1$ or $t_2$ approaches zero. Complete dependence is obtained in the limit when $t_1 = t_2 = 1$ and $r$ tends to infinity. Different limits occur when $t_1$ and $t_2$ are fixed and $r$ tends to infinity. The earliest reference to this model appears to be Joe (1990), who introduces a multivariate extreme value distribution which reduces to $G(z_1,z_2)$ in the bivariate case.

References

Joe, H. (1990) Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab. Letters, 9, 75--81.

See Also

abvneglog, rbvneglog, rgev

Examples

dbvaneglog(matrix(rep(0:4,2),ncol=2), 1.2, c(0.5,1))
pbvaneglog(matrix(rep(0:4,2),ncol=2), 1.2, c(0.5,1))  
rbvaneglog(10, 1.2, c(0.5,1))