Description Usage Arguments Details Value Note References See Also Examples
Density function, distribution function and random generation for the multivariate logistic and multivariate asymmetric logistic models.
1 2 3 4 5 
x, q 
A vector of length 
n 
Number of observations. 
dep 
The dependence parameter(s). For the logistic model, should be a single value. For the asymmetric logistic model, should be a vector of length 2^dd1, or a single value, in which case the value is used for each of the 2^dd1 parameters (see Details). 
asy 
The asymmetry parameters for the asymmetric logistic model. Should be a list with 2^d1 vector elements containing the asymmetry parameters for each separate component (see Details). 
model 
The specified model; a character string. Must be either

d 
The dimension. 
mar 
A vector of length three containing marginal parameters
for every univariate margin, or a matrix with three columns where
each column represents a vector of values to be passed to the
corresponding marginal parameter. It can also be a list with

log 
Logical; if 
lower.tail 
Logical; if 
Define
yi = yi(zi) = {1+si(ziai)/bi}^(1/si)
for 1+si(ziai)/bi > 0 and i = 1,…,d, where the marginal parameters are given by (ai,bi,si), bi > 0. If si = 0 then yi is defined by continuity. Let z = (z1,z2,…,zd). In each of the multivariate distributions functions G(z) given below, the univariate margins are generalized extreme value, so that G(zi) = \exp(yi) for i = 1,…,d. If 1+si(ziai)/bi <= 0 for some i = 1,…,d, 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 d
dimensional multivariate logistic distribution
function with parameter \code{dep} = r is
G(z) = exp{[sum_{i=1}^d yi^(1/r)]^r}
where 0 < r <= 1. This is a special case of the multivariate asymmetric logistic model.
model = "alog"
(Tawn, 1990)
Let B be the set of all nonempty subsets of
{1,…,d}, let
B1={b in B:b=1}, where b
denotes the number of elements in the set b, and let
B(i)={b in B:i in b}.
The d
dimensional multivariate asymmetric logistic distribution
function is
G(z) = exp{sum{b in B} [sum{i in b}(t{i,b}yi)^(1/r{b})]^r{b}},
where the dependence parameters r{b} in (0,1] for all b in B\B1, and the asymmetry parameters t{i,b} in [0,1] for all b in B and i in b. The constraints sum{b in B(i)} t{i,b}=1 for i = 1,…,d ensure that the marginal distributions are generalized extreme value. Further constraints arise from the possible redundancy of asymmetry parameters in the expansion of the distribution form. Let b_{i0} = {i in b:i is not i_0}. If r{b} = 1 for some b in B\B1 then t{i,b} = 0 for all i in b. Furthermore, if for some b in B\B1, t{i,b} = 0 for all i in b_{i0}, then t{i0,b} = 0.
dep
should be a vector of length 2^dd1 which contains
{r{b}:b in B\B1}, with
the order defined by the natural set ordering on the index.
For example, for the trivariate model,
\code{dep} = (r{12},r{13},r{23},r{123}).
asy
should be a list with 2^d1 elements.
Each element is a vector which corresponds to a set
b in B, containing t{i,b} for
every integer i in b.
The elements should be given using the natural set ordering on the
b in B, so that the first d elements are vectors
of length one corresponding to the sets
{1},…,{d}, and the last element is a
a vector of length d, corresponding to the set
{1,…,d}.
asy
must be constructed to ensure that all constraints are
satisfied or an error will occur.
pmvevd
gives the distribution function, dmvevd
gives
the density function and rmvevd
generates random deviates, for
the multivariate logistic or multivariate asymmetric logistic model.
Multivariate extensions of other bivariate models are more complex. A multivariate extension of the HuslerReiss model exists, involving a multidimensional integral and one parameter for each bivariate margin. Multivariate extensions for the negative logistic model can be derived but are considerably more complex and appear to be less flexible. The “multivariate negative logistic model” often presented in the literature (e.g. Kotz et al, 2000) is not a valid distribution function and should not be used.
The logistic and asymmetric logistic models respectively are simulated using Algorithms 2.1 and 2.2 in Stephenson(2003b).
The density function of the logistic model is evaluated using the representation of Shi(1995). The density function of the asymmetric logistic model is evaluated using the representation given in Stephenson(2003a).
Gumbel, E. J. (1960) Distributions des valeurs extremes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris, 9, 171–173.
Kotz, S. and Balakrishnan, N. and Johnson, N. L. (2000) Continuous Multivariate Distributions, vol. 1. New York: John Wiley & Sons, 2nd edn.
Shi, D. (1995) Fisher information for a multivariate extreme value distribution. Biometrika, 82(3), 644–649.
Stephenson, A. G. (2003a) Extreme Value Distributions and their Application. Ph.D. Thesis, Lancaster University, Lancaster, UK.
Stephenson, A. G. (2003b) Simulating multivariate extreme value distributions of logistic type. Extremes, 6(1), 49–60.
Tawn, J. A. (1990) Modelling multivariate extreme value distributions. Biometrika, 77, 245–253.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  pmvevd(matrix(rep(0:4,5), ncol=5), dep = .7, model = "log", d = 5)
pmvevd(rep(4,5), dep = .7, model = "log", d = 5)
rmvevd(10, dep = .7, model = "log", d = 5)
dmvevd(rep(1,20), dep = .7, model = "log", d = 20, log = TRUE)
asy < list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2))
pmvevd(rep(2,3), dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3)
asy < list(.4, .0, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.4,.2))
rmvevd(10, dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3)
dmvevd(rep(0,3), dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3)
asy < list(0, 0, 0, 0, c(0,0), c(0,0), c(0,0), c(0,0), c(0,0), c(0,0),
c(.2,.1,.2), c(.1,.1,.2), c(.3,.4,.1), c(.2,.2,.2), c(.4,.6,.2,.5))
rmvevd(10, dep = .7, asy = asy, model = "alog", d = 4)
rmvevd(10, dep = c(rep(1,6), rep(.7,5)), asy = asy, model = "alog", d = 4)

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