stabtaildepf {copBasic} | R Documentation |
Estimation of the Stable Tail Dependence Function
Description
Kiriliouk et al. (2016, pp. 364–366) describe a technique for estimation of a empirical stable tail dependence function for a random sample. The function is defined as
where is an indicator function,
denotes the
rank()
of the elements and and
is intended to be “large enough” that
has converged to a limit.
The “Capéraà–Fougères smooth” of the empirical stable tail dependence function is defined for a coordinate pair as
where are the weights for the maximum Euclidean likelihood estimator (see
spectralmeas
) and are the pseudo-polar angles (see
spectralmeas
) for the index set defined by
, where
denotes the
-th largest observation of the pseudo-polar radii
where the cardinality of
is exactly
elements long. (Tentatively, then this definition of
is ever so slightly different than in
spectralmeas
.) Lastly, see the multiplier of on the smooth form, and this multiplier is missing in Kiriliouk et al. (2016, p. 365) but shown in Kiriliouk et al. (2016, eq. 17.14, p. 360). Numerical experiments indicate that the
is needed for
but evidently not in
.
The visualization of commences by setting a constant (
) as
(say). The
are solved for
through the
for each of the
. Each solution set constitutes a level set for the stable tail dependence function. If the bivariate data have asymptotic independence (to the right), then a level set or the level sets for all the
are equal to the lines
. Conversely, if the bivariate data have asymptotic dependence (to the right), then the level sets will make 90-degree bends for
.
Usage
stabtaildepf(uv=NULL, xy=NULL, k=function(n) as.integer(0.5*n), levelset=TRUE,
ploton=TRUE, title=TRUE, delu=0.01, smooth=FALSE, ...)
Arguments
uv |
An R |
xy |
A vector of the scalar coordinates |
k |
The |
levelset |
A logical triggering the construction of the level sets for |
ploton |
A logical to call the |
title |
A logical to trigger a title for the plot if |
delu |
The |
smooth |
A logical controlling whether |
... |
Additional arguments to pass. |
Value
Varies according to argument settings. In particular, the levelset=TRUE
will cause an R list
to be returned with the elements having the character string of the respective values and the each holding a
data.frame
of the coordinates.
Note
This function is also called in a secondary recursion mode. The default levelset=TRUE
makes a secondary call with levelset=FALSE
to compute the for the benefit of the looping on the one-dimensional root to solve for a single
in
given a single
. If
levelset=TRUE
and smooth=TRUE
, then a secondary call with smooth=TRUE
and levelset=FALSE
is made to internally return an R list
containing scalar and vectors
and
for similar looping and one-dimensional rooting for
.
If levelset=FALSE
, then xy
is required to hold the coordinate pair of interest. A demonstration follows and shows the limiting behavior of a random sample from the
N4212cop
copula.
n <- 2000 # very CPU intensive this and the next code snippet UV <- simCOP(n=n, cop=N4212cop, para=pi); k <- 1:n lhat <- sapply(k, function(j) stabtaildepf(xy=c(0.1, 0.1), uv=UV, levelset=FALSE, k=j)) plot(k, lhat, xlab="k in [1,n]", cex=0.8, lwd=0.8, type="b", ylab="Empirical Stable Tail Dependence Function") mtext("Empirical function in the 0.10 x 0.10 Pr square (upper left corner)")
The R list
that can be used to compute is retrievable by
x <- 0.1; y <- 0.1; k <- 1:(n-1) lhatCF <- sapply(k, function(j) { Hlis <- stabtaildepf(xy=NA, uv=UV, levelset=FALSE, smooth=TRUE, k=j) 2*sum(Hlis$p3 * sapply(1:Hlis$Nn, function(i) { max(c(Hlis$Wn[i]*x, (1-Hlis$Wn[i])*y)) })) }) lines(k, lhatCF, col="red")
The smooth line (red) of lhatCF
is somewhat closer to the limiting behavior of lhat
, but it is problematic to determine computational consistency. Mathematical consistency with Kiriliouk et al. (2016) appears to be achieved. The Examples section TODO.
Author(s)
William Asquith william.asquith@ttu.edu
References
Beirlant, J., Escobar-Bach, M., Goegebeur, Y., Guillou, A., 2016, Bias-corrected estimation of stable tail dependence function: Journal Multivariate Analysis, v. 143, pp. 453–466, doi:10.1016/j.jmva.2015.10.006.
Kiriliouk, Anna, Segers, Johan, Warchoł, Michał, 2016, Nonparameteric estimation of extremal dependence: in Extreme Value Modeling and Risk Analysis, D.K. Dey and Jun Yan eds., Boca Raton, FL, CRC Press, ISBN 978–1–4987–0129–7.
See Also
Examples
## Not run:
UV <- simCOP(n=1200, cop=GLcop, para=2.1) # Galambos copula
tmp1 <- stabtaildepf(UV) # the lines are curves (strong tail dependence)
tmp2 <- stabtaildepf(UV, smooth=TRUE, ploton=FALSE, col="red") #
## End(Not run)