abvnonpar  R Documentation 
Calculate or plot nonparametric estimates for the dependence function A of the bivariate extreme value distribution.
abvnonpar(x = 0.5, data, epmar = FALSE, nsloc1 = NULL, nsloc2 = NULL, method = c("cfg", "pickands", "tdo", "pot"), k = nrow(data)/4, convex = FALSE, rev = FALSE, madj = 0, kmar = NULL, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0, 1), ylim = c(0.5, 1), xlab = "t", ylab = "A(t)", ...)
x 
A vector of values at which the dependence function is
evaluated (ignored if plot or add is 
data 
A matrix or data frame with two columns, which may contain missing values. 
epmar 
If 
nsloc1, nsloc2 
A data frame with the same number of rows as

method 
The estimation method (see Details). Typically
either 
k 
An integer parameter for the 
convex 
Logical; take the convex minorant? 
rev 
Logical; reverse the dependence function? This is
equivalent to evaluating the function at 
madj 
Performs marginal adjustments for the 
kmar 
In the rare case that the marginal distributions are known, specifies the GEV parameters to be used instead of maximum likelihood estimates. 
plot 
Logical; if 
add 
Logical; add to an existing plot? The existing plot
should have been created using either 
lty, blty 
Function and border line types. Set 
lwd, blwd 
Function and border line widths. 
col 
Line colour. 
xlim, ylim 
x and yaxis limits. 
xlab, ylab 
x and yaxis labels. 
... 
Other highlevel graphics parameters to be passed to

The dependence function A() of the bivariate
extreme value distribution is defined in abvevd
.
Nonparametric estimates are constructed as follows.
Suppose (z_{i1},z_{i2}) for i=1,…,n are n
bivariate observations that are passed using the data
argument.
If epmar
is FALSE
(the default), then
the marginal parameters of the GEV margins are estimated
(under the assumption of independence) and the data is
transformed using
y_{i1} = {1 + s'_1(z_{i1}a'_1)/b'_1}^(1/s'_1)
and
y_{i2} = {1 + s'_2(z_{i2}a'_2)/b'_2}^(1/s'_2)
for i = 1,…,n, where
(a'_1,b'_1,s'_1) and
(a'_2,b'_2,s'_2)
are the maximum likelihood estimates for the location, scale
and shape parameters on the first and second margins.
If nsloc1
or nsloc2
are given, the location
parameters may depend on i (see fgev
).
Two different estimators of the dependence function can be implemented. They are defined (on 0 <= w <= 1) as follows.
method = "cfg"
(Caperaa, Fougeres and Genest, 1997)
log(A_c(w)) = 1/n { sum_{i=1}^n log (max[(1w)y_{i1}, wy_{i1}])  (1w)sum_{i=1}^n y_{i1}  w sum_{i=1}^n y_{i2} }
method = "pickands"
(Pickands, 1981)
A_p(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}/(1w)]}
Two variations on the estimator A_p() are
also implemented. If the argument madj = 1
, an adjustment
given in Deheuvels (1991) is applied. If the argument
madj = 2
, an adjustment given in Hall and Tajvidi (2000)
is applied. These are marginal adjustments; they are only
useful when empirical marginal estimation is used.
Let A_n() be any estimator of A(). None of the estimators satisfy max(w,1w) <= A_n(w) <= 1 for all 0 <= w <= 1. An obvious modification is
A'_n(w) = min(1, max{A_n(w), w, 1w}).
This modification is always implemented.
Convex estimators can be derived by taking the convex minorant,
which can be achieved by setting convex
to TRUE
.
abvnonpar
calculates or plots a nonparametric estimate of
the dependence function of the bivariate extreme value distribution.
I have been asked to point out that Hall and Tajvidi (2000) suggest putting a constrained smoothing spline on their modified Pickands estimator, but this is not done here.
Caperaa, P. Fougeres, A.L. and Genest, C. (1997) A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.
Pickands, J. (1981) Multivariate extreme value distributions. Proc. 43rd Sess. Int. Statist. Inst., 49, 859–878.
Deheuvels, P. (1991) On the limiting behaviour of the Pickands estimator for bivariate extremevalue distributions. Statist. Probab. Letters, 12, 429–439.
Hall, P. and Tajvidi, N. (2000) Distribution and dependencefunction estimation for bivariate extremevalue distributions. Bernoulli, 6, 835–844.
abvevd
, amvnonpar
,
bvtcplot
, fgev
bvdata < rbvevd(100, dep = 0.7, model = "log") abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE) abvnonpar(data = bvdata, method = "pick", plot = TRUE) M1 < fitted(fbvevd(bvdata, model = "log")) abvevd(dep = M1["dep"], model = "log", plot = TRUE) abvnonpar(data = bvdata, add = TRUE, lty = 2)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.