abvnonpar(x = 0.5, data, nsloc1 = NULL, nsloc2 = NULL,
method = c("cfg", "deheuvels", "pickands"), modify = 0,
wf = function(t) t, plot = FALSE, border = TRUE, add = FALSE,
lty = 1, blty = 3, xlim = c(0, 1), ylim = c(0.5, 1),
xlab = "", ylab = "", ...)TRUE).x, for linear modelling of the location
parameter on the first/second margin (see Details).
The data frames are treated as covariate matrices (excluding the
intTRUE the function is plotted and
the values used to create the plot are returned invisibly.TRUE a border representing the
maximal domain is added to the plot.plot.abvnonpar gives a non-parametric estimate of the dependence
function.$A(\cdot)$ is called (by some authors) the dependence function. It follows that $A(0) = A(1) = 1$, and that $A(\cdot)$ is a convex function with $\max(w,1-w) \leq A(w)\leq 1$ for all $0\leq w\leq1$. $A(\cdot)$ does not depend on the marginal parameters.
Suppose $(z_{i1},z_{i2})$ for $i=1,\ldots,n$ are $n$
bivariate observations that are passed using the data
argument.
The marginal parameters are estimated (under the assumption of
independence) and the data is transformed using
$$y_{i1} = {1+\hat{s}_1(z_{i1}-\hat{a}_1)/
\hat{b}_1}_{+}^{-1/\hat{s}_1}$$
and
$$y_{i2} = {1+\hat{s}_2(z_{i2}-\hat{a}_2)/
\hat{b}_2}_{+}^{-1/\hat{s}_2}$$
for $i = 1,\ldots,n$, where
$(\hat{a}_1,\hat{b}_1,\hat{s}_1)$ and
$(\hat{a}_2,\hat{b}_2,\hat{s}_2)$
are the maximum likelihood estimates for the location, scale
and shape parameters on the first and second margins.
If non-stationary fitting is implemented using the nsloc1
or nsloc2 arguments (see fgev or
fbvlog) the marginal location parameters may depend
on $i$.
Three different estimators of the dependence function can be implemented. They are defined (on $0 \leq w \leq 1$) as follows.
$\code{method} = ``pickands''$ (Pickands, 1981) $$A_p(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w}, \frac{y_{i2}}{1-w}\right)\right}^{-1}$$
$\code{method} = ``deheuvels''$ (Deheuvels, 1991) $$A_d(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w}, \frac{y_{i2}}{1-w}\right) - w\sum_{i=1}^n y_{i1} - (1-w) \sum_{i=1}^n y_{i2} + n\right}^{-1}$$
$\code{method} = ``cfg''$; The Default Method (Caperaa, Fougeres and Genest, 1997) $$A_c(w) = \exp\left{ {1-p(t)} \int_{0}^{t} \frac{H(x) - x}{x(1-x)} \, \mbox{d}x - p(t) \int_{t}^{1} \frac{H(x) - x}{x(1-x)} \, \mbox{d}x \right}$$
In the estimator $A_c(\cdot)$, $H(x)$ is the
empirical distribution function of $x_1,\ldots,x_n$, where
$x_i = y_{i2} / (y_{i1} + y_{i2})$ for $i = 1,\ldots,n$,
and $p(t)$ is any bounded function on $[0,1]$, which
can be specified using the argument wf.
By default wf is the identity function.
Let $A_n(\cdot)$ be any estimator of $A(\cdot)$. The estimator $A_d(\cdot)$ satisfies $A_n(0) = A_n(1) = 1$. $A_c(\cdot)$ satisfies this constraint when $p(0) = 0$ and $p(1) = 1$.
None of the estimators satisfy $\max(w,1-w) \leq A_n(w) \leq 1$ for all $0\leq w \leq1$. An obvious modification is $$A_n^{'}(w) = \min(1, \max{A_n(w), w, 1-w}).$$
Another estimator $A_n^{''}(w)$ can be derived by
taking the convex hull of $A_n^{'}(w)$.
These modifications can be implemented using the modify
argument.
Set $\code{modify} = 1$ to plot or calculate
$A_n^{'}(w)$.
Set $\code{modify} = 2$ to plot or calculate
$A_n^{''}(w)$.
$A_n(1/2)$ is returned by default since it is often a useful summary of dependence.
Deheuvels, P. (1991) On the limiting behaviour of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Letters, 12, 429--439.
Pickands, J. (1981) Multivariate extreme value distributions. Proc. 43rd Sess. Int. Statist. Inst., 49, 859--878.
abvlog, fbvlog, fgevbvdata <- rbvlog(100, dep = 0.7)
abvnonpar(seq(0, 1, length = 10), data = bvdata, modify = 2)
abvnonpar(data = bvdata, method = "d", plot = TRUE)Run the code above in your browser using DataLab