evd (version 1.2-3)

bvct: The Bivariate Coles-Tawn Model

Description

Density, distribution function and random generation for the bivariate Coles-Tawn model (also known as the bivariate Dirichelet model).

Usage

dbvct(x, alpha, beta, mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) 
pbvct(q, alpha, beta, mar1 = c(0, 1, 0), mar2 = mar1) 
rbvct(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

• dbvct gives the density, pbvct gives the distribution function and rbvct generates random deviates.

Details

The Coles-Tawn (Coles and Tawn, 1991) distribution function with parameters $\code{alpha} = \alpha > 0$ and $\code{beta} = \beta > 0$ is $$G(z_1,z_2) = \exp\left{-y_1 [1 - \mbox{Be}(q;\alpha+1,\beta)] - y_2 \mbox{Be}(q;\alpha,\beta+1) \right}$$ where $q = \alpha y_2 / (\alpha y_2 + \beta y_1)$ and $\mbox{Be}(q;\alpha,\beta)$ is the beta distribution function evaluated at $q$ with $\code{shape1} = \alpha$, $\code{shape2} = \beta$, and where $$y_i = {1+s_i(z_i-a_i)/b_i}^{-1/s_i}$$ 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.

Complete dependence is obtained in the limit as $\alpha = \beta$ tends to infinity. Independence is obtained as $\alpha = \beta$ approaches zero, and when one of $\alpha,\beta$ is fixed and the other approaches zero. Different limits occur when one of $\alpha,\beta$ is fixed and the other tends to infinity.

References

Coles, S. G. and Tawn, J. A. (1991) Modelling extreme multivariate events. J. Roy. Statist. Soc., B, 53, 377--392.

See Also

abvct, rgev

Examples

dbvct(matrix(rep(0:4,2),ncol=2), .7, 0.52)
pbvct(matrix(rep(0:4,2),ncol=2), .7, 0.52)  
rbvct(10, .7, 0.52)