evd (version 2.1-0)

mvevd: 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 $2^d-d-1$ paramete
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 wit
log
Logical; if TRUE, the log density is returned.
lower.tail
Logical; if TRUE (default), probabilities are P[X <= x],="" otherwise,="" p[x=""> x]

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.

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 $i$th 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.

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

Run this code
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)

Run the code above in your browser using DataCamp Workspace