evd (version 1.2-3)

bvalog: The Bivariate Asymmetric Logistic Model

Description

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

Usage

dbvalog(x, dep, asy = c(1, 1), mar1 = c(0, 1, 0), mar2 = mar1,
    log = FALSE) 
pbvalog(q, dep, asy = c(1, 1), mar1 = c(0, 1, 0), mar2 = mar1) 
rbvalog(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

• dbvalog gives the density, pbvalog gives the distribution function and rbvalog generates random deviates.

Details

The bivariate asymmetric logistic distribution function with parameters $\code{dep} = r$ and $\code{asy} = (t_1,t_2)$ is $$G(z_1,z_2) = \exp\left{-(1-t_1)y_1-(1-t_2)y_2- [(t_1y_1)^{1/r}+(t_2y_2)^{1/r}]^r\right}$$ where $0 < r \leq 1$, $0 \leq t_1,t_2 \leq 1$, 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 logistic model is equivalent to the logistic model. Independence is obtained when either $r = 1$, $t_1 = 0$ or $t_2 = 0$. Complete dependence is obtained in the limit when $t_1 = t_2 = 1$ and $r$ approaches zero. Different limits occur when $t_1$ and $t_2$ are fixed and $r$ approaches zero. The model was first introduced by Tawn (1988).

References

Stephenson, A. G. (2002) Simulating multivariate extreme value distributions of logistic type. To be published - available on request.

Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397--415.

See Also

abvlog, rbvlog, rgev, rmvalog

Examples

dbvalog(matrix(rep(0:4,2),ncol=2), .7, c(0.5,1))
pbvalog(matrix(rep(0:4,2),ncol=2), .7, c(0.5,1))  
rbvalog(10, .7, c(0.5,1))