evd (version 1.2-3)

bvlog: The Bivariate Logistic Model

Description

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

Usage

dbvlog(x, dep, mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) 
pbvlog(q, dep, mar1 = c(0, 1, 0), mar2 = mar1) 
rbvlog(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

• dbvlog gives the density, pbvlog gives the distribution function and rbvlog generates random deviates.

Details

The bivariate logistic distribution function (Gumbel, 1960) with parameter $\code{dep} = r$ is $$G(z_1,z_2) = \exp\left[-(y_1^{1/r}+y_2^{1/r})^r\right]$$ where $0 < r \leq 1$ 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. This is a special case of the bivariate asymmetric logistic model. The univariate marginal distributions are generalized extreme value.

Complete dependence is obtained in the limit as $r$ approaches zero. Independence is obtained when $r = 1$.

References

Gumbel, E. J. (1960) Distributions des valeurs extremes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris, 9, 171--173. Stephenson, A. G. (2002) Simulating multivariate extreme value distributions of logistic type. To be published - available on request.

See Also

abvlog, rbvalog, rgev, rmvlog

Examples

dbvlog(matrix(rep(0:4,2),ncol=2), .7)
pbvlog(matrix(rep(0:4,2),ncol=2), .7)  
rbvlog(10, .7)