abvnonpar(x = 0.5, data, nsloc1 = NULL, nsloc2 = NULL,
method = c("cfg", "deheuvels", "pickands"), modify = 0,
wf = function(t) t, plot = FALSE, add = FALSE, lty = 1, lwd = 1,
col = 1, blty = 3, xlim = c(0, 1), ylim = c(0.5, 1), xlab = "",
ylab = "", ...)TRUE).data, for linear modelling of the location
parameter on the first/second margin (see Details).
The data frames are treated as covariate matrices, excluding the
intercept.
TRUE the function is plotted and
the values used to create the plot are returned invisibly.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 nsloc1 or nsloc2 are given, the location
parameters may depend on $i$ (see fgev
for details).
Three different estimators of the dependence function can be implemented. They are defined (on $0 \leq w \leq 1$) as follows.
method = "pickands" (Pickands, 1981)
$$A_p(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w},
\frac{y_{i2}}{1-w}\right)\right}^{-1}$$
method = "deheuvels" (Deheuvels, 1991)
$$A_d(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w},
\frac{y_{i2}}{1-w}\right) - w\sum_{i=1}^n y_{i1} - (1-w)
\sum_{i=1}^n y_{i2} + n\right}^{-1}$$
method = "cfg"; The Default Method
(Caperaa, Fougeres and Genest, 1997)
$$A_c(w) = \exp\left{ {1-p(w)} \int_{0}^{w}
\frac{H(x) - x}{x(1-x)} \, \mbox{d}x - p(w) \int_{w}^{1}
\frac{H(x) - x}{x(1-x)} \, \mbox{d}x \right}$$
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 $p(w)$ 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, bvdp,
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