dbvevd(x, dep, asy = c(1, 1), alpha, beta, model = "log",
mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE)
pbvevd(q, dep, asy = c(1, 1), alpha, beta, model = "log",
mar1 = c(0, 1, 0), mar2 = mar1, lower.tail = TRUE)
rbvevd(n, dep, asy = c(1, 1), alpha, beta, model = "log",
mar1 = c(0, 1, 0), mar2 = mar1)"log" (the default), "alog", "hr",
"neglog", "aneglog", "bilog",
"negbilog" or "ct"TRUE, the log density is returned.TRUE (default), probabilities
are P[X <= x],="" otherwise,="" p[x=""> x].dbvevd gives the density function, pbvevd gives the
distribution function and rbvevd generates random deviates,
for one of eight parametric bivariate extreme value models.model = "log" (Gumbel, 1960)
The bivariate logistic distribution function with
parameter $\code{dep} = r$ is
$$G(z_1,z_2) = \exp\left[-(y_1^{1/r}+y_2^{1/r})^r\right]$$
where $0 < r \leq 1$.
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
$\code{asy} = (t_1,t_2)$ is
$$G(z_1,z_2) = \exp\left{-(1-t_1)y_1-(1-t_2)y_2-
[(t_1y_1)^{1/r}+(t_2y_2)^{1/r}]^r\right}$$
where $0 < r \leq 1$ and
$0 \leq t_1,t_2 \leq 1$.
When $t_1 = t_2 = 1$ the asymmetric logistic
model is equivalent to the logistic model.
Independence is obtained when either $r = 1$,
$t_1 = 0$ or $t_2 = 0$.
Complete dependence is obtained in the limit when
$t_1 = t_2 = 1$ and $r$
approaches zero.
Different limits occur when $t_1$ and $t_2$
are fixed and $r$ approaches zero.
model = "hr" (Husler and Reiss, 1989)
The Husler-Reiss distribution function with parameter
$\code{dep} = r$ is
$$G(z_1,z_2) = \exp\left(-y_1\Phi{r^{-1}+{\textstyle\frac{1}{2}}
r[\log(y_1/y_2)]} - y_2\Phi{r^{-1}+{\textstyle\frac{1}{2}}r
[\log(y_2/y_1)]}\right)$$
where $\Phi(\cdot)$ 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 $$G(z_1,z_2) = \exp\left{-y_1-y_2+ [y_1^{-r}+y_2^{-r}]^{-1/r}\right}$$ 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
$\code{asy} = (t_1,t_2)$ is
$$G(z_1,z_2) = \exp\left{-y_1-y_2+
[(t_1y_1)^{-r}+(t_2y_2)^{-r}]^{-1/r}\right}$$
where $r > 0$ and $0 < t_1,t_2 \leq 1$.
When $t_1 = t_2 = 1$ the asymmetric negative
logistic model is equivalent to the negative logistic model.
Independence is obtained in the limit as either $r$,
$t_1$ or $t_2$ approaches zero.
Complete dependence is obtained in the limit when
$t_1 = t_2 = 1$ and $r$
tends to infinity.
Different limits occur when $t_1$ and $t_2$
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 $G(z_1,z_2)$ in the bivariate case.
model = "bilog" (Smith, 1990)
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
$$(1-\alpha) y_1 (1-q)^\beta - (1-\beta) y_2 q^\alpha = 0,$$
$0 < \alpha,\beta < 1$.
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 $$G(z_1,z_2) = \exp\left{- y_1 - y_2 + y_1 q^{1+\alpha} + y_2 (1-q)^{1+\beta}\right}$$ where $q = q(y_1,y_2;\alpha,\beta)$ is the root of the equation $$(1+\alpha) y_1 q^\alpha - (1+\beta) y_2 (1-q)^\beta = 0,$$ $\alpha > 0$ 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
$$G(z_1,z_2) =
\exp\left{-y_1 [1 - \mbox{Be}(q;\alpha+1,\beta)] -
y_2 \mbox{Be}(q;\alpha,\beta+1) \right}$$
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
$\code{shape1} = \alpha$ 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.
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. To be published - available on request.
Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397--415.
abvpar, rgev, rmvevdpbvevd(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")Run the code above in your browser using DataLab