evd (version 1.2-3)

mvalog: The Multivariate Asymmetric Logistic Model

Description

Distribution function and random generation for the multivariate asymmetric logistic model.

Usage

pmvalog(q, dep, asy, d = 2, mar = c(0, 1, 0)) 
rmvalog(n, dep, asy, d = 2, mar = c(0, 1, 0))

Arguments

q

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

n

Number of observations.

dep

A vector of length $2^d-d-1$ containing the dependence parameters (see Details).

asy

A list with $2^d-1$ vector elements containing asymmetry parameters for each separate component (see Details).

d

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 on every univariate margin

Value

• pmvalog gives the distribution function and rmvalog generates random deviates.

Details

Let $z = (z_1,z_2,\ldots,z_d)$. Let $B$ be the set of all non-empty subsets of ${1,\ldots,d}$, let $B_1={b \in B:|b|=1}$ 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 $i\in b$, and where $$y_i = {1+s_i(z_i-a_i)/b_i}^{-1/s_i}$$ for $1+s_i(z_i-a_i)/b_i > 0$ and $i = 1,\ldots,d$. If $s_i = 0$ then $y_i$ is defined by continuity. The univariate marginal distributions are generalized extreme value. The model was first introduced in full generality by Tawn (1990). Different parameters on each margin are not implemented, so $\code{mar} = (a_i,b_i,s_i)$ for every $i$. 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. More subtle constraints exist and can be found in the user's guide.

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. In particular, for the trivariate model dep should be given as $(r_{12},r_{13},r_{23},r_{123})$. 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. See the user's guide for detailed examples.

References

Kotz, S. and Balakrishnan, N. and Johnson, N. L. (2000) Continuous Multivariate Distributions, vol. 1. New York: John Wiley & Sons, 2nd edn. Stephenson, A. G. (2002) Simulating multivariate extreme value distributions of logistic type. To be published - available on request.

Tawn, J. A. (1990) Modelling multivariate extreme value distributions. Biometrika, 77, 245--253.

See Also

rbvalog, rmvlog, rgev

Examples

pmvalog(c(2, 2, 2), dep = c(.6,.5,.8,.3),
  asy = list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2)),
  d = 3)
rmvalog(3, dep = c(.6,.5,.8,.3),
  asy = list(.4, .0, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.4,.2)),
  d = 3)
set.seed(50)
rmvalog(3, dep = rep(.7,11),
  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)),
  d = 4)
set.seed(50)
rmvalog(3, dep = c(rep(1,6),rep(.7,5)),
  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)),
  d = 4)