evd (version 1.2-3)

bvbilog: The Bivariate Bilogistic Model

Description

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

Usage

dbvbilog(x, alpha, beta, mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) 
pbvbilog(q, alpha, beta, mar1 = c(0, 1, 0), mar2 = mar1) 
rbvbilog(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

• dbvbilog gives the density, pbvbilog gives the distribution function and rbvbilog generates random deviates.

Details

The bilogistic distribution function with parameters $\code{alpha} = \alpha$ and $\code{beta} = \beta$ is $$G(z_1,z_2) = \exp\left{-y_1 q^{1-\alpha} - y_2 (1-q)^{1-\beta}\right}$$ where $q = q(y_1,y_2;\alpha,\beta)$ is the root of the equation $$(1-\alpha) y_1 (1-q)^\beta - (1-\beta) y_2 q^\alpha = 0,$$ $0 < \alpha,\beta < 1$ 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 $\alpha = \beta$ the bilogistic model is equivalent to the logistic model with dependence parameter $\code{dep} = \alpha = \beta$. Complete dependence is obtained in the limit as $\alpha = \beta$ approaches zero. Independence is obtained as $\alpha = \beta$ approaches one, and when one of $\alpha,\beta$ is fixed and the other approaches one. Different limits occur when one of $\alpha,\beta$ is fixed and the other approaches zero. A bilogistic model is fitted in Smith (1990), where it appears to have been first introduced.

References

Joe, H. (1997) Multivariate Models and Dependence Concepts, London: Chapman & Hall.

Smith, R. L. (1990) Extreme value theory. In Handbook of Applicable Mathematics (ed. W. Ledermann), vol. 7. Chichester: John Wiley, pp. 437--471.

See Also

abvbilog, rbvnegbilog, rgev

Examples

dbvbilog(matrix(rep(0:4,2),ncol=2), .7, 0.52)
pbvbilog(matrix(rep(0:4,2),ncol=2), .7, 0.52)  
rbvbilog(10, .7, 0.52)