mvevd {evd}

Parametric Multivariate Extreme Value Distributions

Description

Density function, distribution function and random generation for the multivariate logistic and multivariate asymmetric logistic models.

Usage

pmvevd(q, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0),
    lower.tail = TRUE)
rmvevd(n, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0))
dmvevd(x, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0),
    log = FALSE) 

Arguments

x, q

A vector of length d or a matrix with d columns, in which case the density/distribution is evaluated across the rows.

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^d-d-1, or a single value, in which case the value is used for each of the 2^d-d-1 parameters (see Details).

asy

The asymmetry parameters for the asymmetric logistic model. Should be a list with 2^d-1 vector elements containing the asymmetry parameters for each separate component (see Details).

model

The specified model; a character string. Must be either "log" (the default) or "alog" (or any unique partial match), for the logistic and asymmetric logistic models respectively.

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 d elements, such that each element is either a vector of length three or a matrix with three columns, in which case the ith element represents the marginal parameters on the ith margin.

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

y_i = y_i(z_i) = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}

for 1+s_i(z_i-a_i)/b_i > 0 and i = 1,\ldots,d, where the marginal parameters are given by (a_i,b_i,s_i), b_i > 0. If s_i = 0 then y_i is defined by continuity. Let z = (z_1,z_2,\ldots,z_d). In each of the multivariate distributions functions G(z) given below, the univariate margins are generalized extreme value, so that G(z_i) = \exp(-y_i) for i = 1,\ldots,d. If 1+s_i(z_i-a_i)/b_i \leq 0 for some i = 1,\ldots,d, the value z_i is either greater than the upper end point (if s_i < 0), or less than the lower end point (if s_i > 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\left\{-\left(\sum\nolimits_{i = 1}^{d} y_i^{1/r}\right)^r\right\}

where 0 < r \leq 1. This is a special case of the multivariate asymmetric logistic model.

model = "alog" (Tawn, 1990)

Let B be the set of all non-empty subsets of \{1,\ldots,d\}, let B_1=\{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\left\{-\sum\nolimits_{b \in B} \left[\sum\nolimits_ {i\in b}(t_{i,b}y_i)^{1/r_b}\right]^{r_b}\right\},

where the dependence parameters r_b\in(0,1] for all b\in B \setminus B_1, 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,\ldots,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_{-i_0} = \{i \in b:i \neq i_0\}. If r_b = 1 for some b\in B \setminus B_1 then t_{i,b} = 0 for all i\in b. Furthermore, if for some b\in B \setminus B_1, t_{i,b} = 0 for all i\in b_{-i_0}, then t_{i_0,b} = 0.

dep should be a vector of length 2^d-d-1 which contains \{r_b:b\in B \setminus B_1\}, 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^d-1 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\},\ldots,\{d\}, and the last element is a a vector of length d, corresponding to the set \{1,\ldots,d\}. asy must be constructed to ensure that all constraints are satisfied or an error will occur.

Value

pmvevd gives the distribution function, dmvevd gives the density function and rmvevd generates random deviates, for the multivariate logistic or multivariate asymmetric logistic model.

Note

Multivariate extensions of other bivariate models are more complex. A multivariate extension of the Husler-Reiss 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).

References

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.

See Also

rbvevd, rgev

Examples

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)