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"
, "ct"
oTRUE
, the log density is returned.TRUE
(default), the
distribution function is returned; the survivor function
is returned otherwise.dbvevd
gives the density function, pbvevd
gives the
distribution function and rbvevd
generates random deviates,
for one of nine 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.
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.
$$A(t) = 1 - (\alpha +\beta)t + \alpha t^2 + \beta t^3$$
where $\alpha$ and $\alpha + 3\beta$
are non-negative, and where $\alpha + \beta$
and $\alpha + 2\beta$ 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
.
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
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")
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