atvpar: Parametric Dependence Functions of Trivariate Extreme Value Models

Description

Calculate or plot the dependence function $A$ for the trivariate logistic and trivariate asymmetric logistic models.

Usage

atvpar(x = rep(1/3,3), dep, asy, model = c("log", "alog"), plot =
    FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50,
    lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1)

Arguments

A vector of length three or a matrix with three columns, in which case the dependence function is evaluated across the rows (ignored if plot is TRUE). The elements/rows of the vector/matrix should be positive and should sum to one

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 four, or a single value, in which case the value is used for each of the four parameters (see

asy

The asymmetry parameters for the asymmetric logistic model. Should be a list with seven vector elements; three of length one, three of length two and one of length three, containing the asymmetry parameters for each separate component (see

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. The definition of each model is given

plot

Logical; if TRUE the function is plotted. The minimum (evaluated) value is returned invisibly. If FALSE (the default), the following arguments, excluding ord, are ignored.

col

A list of colours (see image). The first colours in the list represent smaller values, and hence stronger dependence. Each colour represents an equally spaced interval between lower a

blty

The border line type, for the border that surrounds the triangular image. By default blty is zero, so no border is plotted. Plotting a border leads to (by default) an increase in grid (and hence computation time), to

grid

For plotting, the function is evaluated at grid^2 points.

lower

The minimum value for which colours are plotted. By defualt $\code{lower} = 1/3$ as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.

ord

A vector of length three, which should be a permutation of the set ${1,2,3}$. For plotting, the points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ (the vertices of the simplex) are depicted clockwise from the top in the order defined by ord

lab

A character vector of length three, in which case the
    ith margin is labelled using the ith component,
    or NULL, in which case no labels are given. The actual
    location of the margins, and hence the labels,

lcex

A numerical value giving the amount by which the
    labels should be scaled relative to the default. Ignored
    if lab is NULL.

`Value`

atvpar calculates or plots the dependence function
  for the trivariate logistic and trivariate asymmetric logistic
  models, at specified parameter values.

`Details`

Let $z = (z_1,z_2,z_3)$ and
  $w = (w_1,w_2,w_3)$.
  Any trivariate extreme value distribution can be written as
  $$G(z) = \exp\left{- \left{\sum\nolimits_{j=1}^{3} y_j
    \right} A\left(\frac{y_1}{\sum\nolimits_{j=1}^{3} y_j},
    \frac{y_2}{\sum\nolimits_{j=1}^{3} y_j},
    \frac{y_3}{\sum\nolimits_{j=1}^{3} y_j}\right)\right}$$
  for some function $A$ defined on the simplex
  $S_3 = {w \in R^3_+ : \sum\nolimits_{j=1}^{3}
    w_j = 1}$,
  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
  $i = 1,2,3$, and where the (generalized extreme value)
  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.
  $A$ is called (by some authors) the dependence function.
  It follows that $A(1,0,0) = A(0,1,0) = A(0,0,1) = 1$, and
  that $A$ is a convex function with
  $\max(w_1,w_2,w_3) \leq A(w)\leq 1$ for
  all $w$ in $S_3$.
  The lower and upper limits of $A$ are obtained under complete
  dependence and mutual independence respectively.
  $A$ does not depend on the marginal parameters.

`See Also`

atvnonpar, abvpar,
  rmvevd, image

`Examples`

Run this codeatvpar(dep = 0.5, model = "log")
s3pts <- matrix(rexp(30), nrow = 10, ncol = 3)
s3pts <- s3pts/rowSums(s3pts)
atvpar(s3pts, dep = 0.5, model = "log")
atvpar(dep = 0.05, model = "log", plot = TRUE, blty = 1)
atvpar(dep = 0.95, model = "log", plot = TRUE, lower = 0.94)

asy <- list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2))
atvpar(s3pts, dep = 0.15, asy = asy, model = "alog")
atvpar(dep = 0.15, asy = asy, model = "al", plot = TRUE, lower = 0.7)
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