# bvevd: Parametric Bivariate Extreme Value Distributions In evd: Functions for Extreme Value Distributions

## Description

Density function, distribution function and random generation for nine parametric bivariate extreme value models.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```dbvevd(x, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) pbvevd(q, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0, 1, 0), mar2 = mar1, lower.tail = TRUE) rbvevd(n, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), 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 for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models. `asy` A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models. `alpha, beta` Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models. `model` The specified model; a character string. Must be either `"log"` (the default), `"alog"`, `"hr"`, `"neglog"`, `"aneglog"`, `"bilog"`, `"negbilog"`, `"ct"` or `"amix"` (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning. `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. `lower.tail` Logical; if `TRUE` (default), the distribution function is returned; the survivor function is returned otherwise.

## Details

Define

yi = yi(zi) = {1+si(zi-ai)/bi}^(-1/si)

for 1+si(zi-ai)/bi > 0 and i = 1,2, where the marginal parameters are given by \code{mari} = (ai,bi,si), bi > 0. If si = 0 then yi is defined by continuity.

In each of the bivariate distributions functions G(z1,z2) given below, the univariate margins are generalized extreme value, so that G(zi) = \exp(-yi) for i = 1,2. If 1+si(zi-ai)/bi <= 0 for some i = 1,2, the value zi is either greater than the upper end point (if si < 0), or less than the lower end point (if si > 0), of the ith univariate marginal distribution.

`model = "log"` (Gumbel, 1960)

The bivariate logistic distribution function with parameter \code{dep} = r is

G(z1,z2) = exp{-[y1^(1/r)+y2^(1/r)]^r}

where 0 < r <= 1. This is a special case of the bivariate asymmetric logistic model. Complete dependence is obtained in the limit as r approaches zero. Independence is obtained when r = 1.

`model = "alog"` (Tawn, 1988)

The bivariate asymmetric logistic distribution function with parameters \code{dep} = r and \code{asy} = (t1,t2) is

G(z1,z2) = exp{-(1-t1)y1-(1-t2)y2-[(t1y1)^(1/r)+(t2y2)^(1/r)]^r}

where 0 < r <= 1 and 0 <= t1,t2 <= 1. When t1 = t2 = 1 the asymmetric logistic model is equivalent to the logistic model. Independence is obtained when either r = 1, t1 = 0 or t2 = 0. Complete dependence is obtained in the limit when t1 = t2 = 1 and r approaches zero. Different limits occur when t1 and t2 are fixed and r approaches zero.

`model = "hr"` (Husler and Reiss, 1989)

The Husler-Reiss distribution function with parameter \code{dep} = r is

G(z1,z2) = exp(-y1 Phi{r^{-1}+r[log(y1/y2)]/2} - y2 Phi{r^{-1}+r[log(y2/y1)]/2}

where Phi() is the standard normal distribution function and r > 0. Independence is obtained in the limit as r approaches zero. Complete dependence is obtained as r tends to infinity.

`model = "neglog"` (Galambos, 1975)

The bivariate negative logistic distribution function with parameter \code{dep} = r is

G(z1,z2) = exp{-y1-y2+[y1^(-r)+y2^(-r)]^(-1/r)}

where r > 0. This is a special case of the bivariate asymmetric negative logistic model. Independence is obtained in the limit as r approaches zero. Complete dependence is obtained as r tends to infinity. The earliest reference to this model appears to be Galambos (1975, Section 4).

`model = "aneglog"` (Joe, 1990)

The bivariate asymmetric negative logistic distribution function with parameters parameters \code{dep} = r and \code{asy} = (t1,t2) is

G(z1,z2) = exp{-y1-y2+[(t1y1)^(-r)+(t2y2)^(-r)]^(-1/r)}

where r > 0 and 0 < t1,t2 <= 1. When t1 = t2 = 1 the asymmetric negative logistic model is equivalent to the negative logistic model. Independence is obtained in the limit as either r, t1 or t2 approaches zero. Complete dependence is obtained in the limit when t1 = t2 = 1 and r tends to infinity. Different limits occur when t1 and t2 are fixed and r tends to infinity. The earliest reference to this model appears to be Joe (1990), who introduces a multivariate extreme value distribution which reduces to G(z1,z2) in the bivariate case.

`model = "bilog"` (Smith, 1990)

The bilogistic distribution function with parameters \code{alpha} = alpha and \code{beta} = beta is

G(z1,z2) = exp{- y1 q^(1-alpha) - y2 (1-q)^(1-beta)}

where q = q(y1,y2;alpha,beta) is the root of the equation

(1-alpha) y1 (1-q)^beta - (1-beta) y2 q^alpha = 0,

0 < alpha,beta < 1. 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.

`model = "negbilog"` (Coles and Tawn, 1994)

The negative bilogistic distribution function with parameters \code{alpha} = alpha and \code{beta} = beta is

G(z1,z2) = exp{- y1 - y2 + y1 q^(1+alpha) + y2 (1-q)^(1+beta)}

where q = q(y1,y2;alpha,beta) is the root of the equation

(1+alpha) y1 q^alpha - (1+beta) y2 (1-q)^beta = 0,

alpha > 0 and beta > 0. 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.

`model = "ct"` (Coles and Tawn, 1991)

The Coles-Tawn distribution function with parameters \code{alpha} = alpha > 0 and \code{beta} = beta > 0 is

G(z1,z2) = exp{- y1 [1 - Be(q;alpha+1,beta)] - y2 Be(q;alpha,beta+1)}

where q = alpha y2 / (alpha y2 + beta y1) and Be(q;alpha,beta) is the beta distribution function evaluated at q with \code{shape1} = alpha and \code{shape2} = beta. 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.

`model = "amix"` (Tawn, 1988)

The asymmetric mixed distribution function with parameters \code{alpha} = alpha and \code{beta} = beta has a dependence function with the following cubic polynomial form.

A(t) = 1 - (α +β)t + α t^2 + β t^3

where alpha and alpha + 3beta are non-negative, and where alpha + beta and alpha + 2beta are less than or equal to one. These constraints imply that beta lies in the interval [-0.5,0.5] and that alpha lies in the interval [0,1.5], though alpha can only be greater than one if beta is negative. The strength of dependence increases for increasing alpha (for fixed beta). Complete dependence cannot be obtained. Independence is obtained when both parameters are zero. For the definition of a dependence function, see `abvevd`.

## Value

`dbvevd` gives the density function, `pbvevd` gives the distribution function and `rbvevd` generates random deviates, for one of nine parametric bivariate extreme value models.

## Note

The logistic and asymmetric logistic models respectively are simulated using bivariate versions of Algorithms 1.1 and 1.2 in Stephenson(2003). All other models are simulated using a root finding algorithm to simulate from the conditional distributions.

The simulation of the bilogistic and negative bilogistic models requires a root finding algorithm to evaluate q within the root finding algorithm used to simulate from the conditional distributions. The generation of bilogistic and negative bilogistic random deviates is therefore relatively slow (about 2.8 seconds per 1000 random vectors on a 450MHz PIII, 512Mb RAM).

The bilogistic and negative bilogistic models can be represented under a single model, using the integral of the maximum of two beta distributions (Joe, 1997).

The Coles-Tawn model is called the Dirichelet model in Coles and Tawn (1991).

## References

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

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.

Galambos, J. (1975) Order statistics of samples from multivariate distributions. J. Amer. Statist. Assoc., 70, 674–680.

Gumbel, E. J. (1960) Distributions des valeurs extremes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris, 9, 171–173.

Husler, J. and Reiss, R.-D. (1989) Maxima of normal random vectors: between independence and complete dependence. Statist. Probab. Letters, 7, 283–286.

Joe, H. (1990) Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab. Letters, 9, 75–81.

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.

Stephenson, A. G. (2003) Simulating multivariate extreme value distributions of logistic type. Extremes, 6(1), 49–60.

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

`abvevd`, `rgev`, `rmvevd`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```pbvevd(matrix(rep(0:4,2), ncol=2), dep = 0.7, model = "log") pbvevd(c(2,2), dep = 0.7, asy = c(0.6,0.8), model = "alog") pbvevd(c(1,1), dep = 1.7, model = "hr") margins <- cbind(0, 1, seq(-0.5,0.5,0.1)) rbvevd(11, dep = 1.7, model = "hr", mar1 = margins) rbvevd(10, dep = 1.2, model = "neglog", mar1 = c(10, 1, 1)) rbvevd(10, alpha = 0.7, beta = 0.52, model = "bilog") dbvevd(c(0,0), dep = 1.2, asy = c(0.5,0.9), model = "aneglog") dbvevd(c(0,0), alpha = 0.75, beta = 0.5, model = "ct", log = TRUE) dbvevd(c(0,0), alpha = 0.7, beta = 1.52, model = "negbilog") ```