stabtaildepf: 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 $$\widehat{l}(x,y) = \frac{1}{k}\sum_{i=1}^n \mathbf{1}\bigl[ R_{i,x,n} > n + 1 - kx \mbox{\ or\ } R_{i,y,n} > n + 1 - ky \bigr]\mbox{,}$$ where $\mathbf{1}[\cdot]$ is an indicator function, $R$ denotes the rank() of the elements and $k \in [1,\ldots,n]$ and $k$ is intended to be “large enough” that $\widehat{l}(x,y)$ has converged to a limit.

The “Cap<U+00E9>ra<U+00E0>--Foug<U+00E8>res smooth” of the empirical stable tail dependence function is defined for a coordinate pair $(x,y)$ as $$\widehat{l}_{CF}(x,y) = 2 \sum_{i \in I_n} \widehat{p}_{3,i} \times \mathrm{max}\bigl[\widehat{W}_i x,\, (1-\widehat{W}_i) y \bigr]\mbox{,}$$ where $\widehat{p}_{3,i}$ are the weights for the maximum Euclidean likelihood estimator (see spectralmeas) and $\widehat{W}_i$ are the pseudo-polar angles (see spectralmeas) for the index set $I_n$ defined by $I_n = \{i = 1, \ldots, n : \widehat{S}_i > \widehat{S}_{(k{+}1)}\}$, where $\widehat{S}_{(k+1)}$ denotes the $(k{+}1)$-th largest observation of the pseudo-polar radii $\widehat{S}_i$ where the cardinality of $I_n$ is exactly $k$ elements long. (Tentatively, then this definition of $I_n$ is ever so slightly different than in spectralmeas.) Lastly, note the multiplier of $2$ 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 $2$ is needed for $\widehat{l}_{CF}(x,y)$ but evidently not in $\widehat{l}(x,y)$.

The visualization of $l(x,y)$ commences by setting a constant ($c > 0$) as $c_i \in 0.2,0.4,0.6,0.8$ (say). The $y$ are solved for $x \in [0,\ldots,c_i]$ through the $l(x,y)$ for each of the $c_i$. 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 $c$ are equal to the lines $x + y = c$. Conversely, if the bivariate data have asymptotic dependence (to the right), then the level sets will make 90-degree bends for $\mathrm{max}(x,y) = c$.

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

An R data.frame of $u$ and $v$ nonexceedance probabilities in the respective $X$ (horizontal) and $Y$ (vertical) directions. Note that rank()s are called on these so strictly speaking this need not be as nonexceedance probabilities. This is not an optional argument;

A vector of the scalar coordinates $(x,y)$, which are “the relative distances to the upper endpoints of [these respective] variables” (Kiriliouk et al., 2016, p. 356). This is a major point of nomenclature confusion. If these are in probability units, they are exceedance probabilities. Though tested for NULL and a warning issued, these can be NULL only if levelset=TRUE but can be set to xy=NA if levelset=FALSE and smooth=TRUE (see discussion in Note);

The $k$ for both the $\widehat{l}(x,y)$ and $\widehat{l}_{CF}$, though the effect of $k$ might not quite be the same for each. The default seems to work fairly well;

levelset

A logical triggering the construction of the level sets for $c$ $=$ seq(0.1,1, by=0.1);

ploton

A logical to call the plot() function;

title

A logical to trigger a title for the plot if ploton=TRUE;

delu

The $\Delta x$ for a sequence of $x$ $=$ seq(0,c,by=delu);

smooth

A logical controlling whether $\widehat{l}(x,y)$ or the Cap<U+00E9>ra<U+00E0>--Foug<U+00E8>res smooth function $\widehat{l}_{CF}(x,y)$ is used;

...

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 $c$ values and the each holding a data.frame of the $(x,y)$ coordinates.

References

Kiriliouk, Anna, Segers, Johan, Warcho<U+0142>, Micha<U+0142>, 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.

Examples

Run this code

# 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=2) # 
# }