evd (version 1.2-3)

bvnegbilog: The Bivariate Negative Bilogistic Model

Description

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

Usage

dbvnegbilog(x, alpha, beta, mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) 
pbvnegbilog(q, alpha, beta, mar1 = c(0, 1, 0), mar2 = mar1) 
rbvnegbilog(n, alpha, beta, 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.

alpha

Alpha parameter.

beta

Beta 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

• dbvnegbilog gives the density, pbvnegbilog gives the distribution function and rbvnegbilog generates random deviates.

Details

The negative bilogistic distribution function with parameters $\code{alpha} = \alpha$ and $\code{beta} = \beta$ is $$G(z_1,z_2) = \exp\left{- y_1 - y_2 + y_1 q^{1+\alpha} + y_2 (1-q)^{1+\beta}\right}$$ where $q = q(y_1,y_2;\alpha,\beta)$ is the root of the equation $$(1+\alpha) y_1 q^\alpha - (1+\beta) y_2 (1-q)^\beta = 0,$$ $\alpha > 0$, $\beta > 0$ 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.

When $\alpha = \beta$ the negative bilogistic model is equivalent to the negative logistic model with dependence parameter $\code{dep} = 1/\alpha = 1/\beta$. Complete dependence is obtained in the limit as $\alpha = \beta$ approaches zero. Independence is obtained as $\alpha = \beta$ tends to infinity, and when one of $\alpha,\beta$ is fixed and the other tends to infinity. Different limits occur when one of $\alpha,\beta$ is fixed and the other approaches zero. The model was first introduced by Coles and Tawn (1994).

References

Coles, S. G. and Tawn, J. A. (1994) Statistical methods for multivariate extremes: an application to structural design (with discussion). Appl. Statist., 43, 1--48. Joe, H. (1997) Multivariate Models and Dependence Concepts, London: Chapman & Hall.

See Also

abvnegbilog, rbvbilog, rgev

Examples

dbvnegbilog(matrix(rep(0:4,2),ncol=2), .7, 1.52)
pbvnegbilog(matrix(rep(0:4,2),ncol=2), .7, 1.52)  
rbvnegbilog(10, .7, 1.52)