# Parametric Bivariate 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 |

`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 |

`lower.tail` |
Logical; if |

### 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 *i*th 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.

### See Also

`abvevd`

, `rgev`

, `rmvevd`

### Examples

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")
``` |

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