abvnonpar(x = 0.5, data, nsloc1 = NULL, nsloc2 = NULL,
method = c("cfg", "pickands", "deheuvels", "halltajvidi", "tdo"),
convex = FALSE, 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). $A(1/2)$
is returned by default since it is often a useful summary of
dependence.data, for linear modelling of the location
parameter on the first/second margin.
The data frames are treated as covariate matrices, excluding the
intercept.
A numeric vector can"cfg" method is used by default."cfg"
method (see Details). The function must be vectorized.TRUE the function is plotted. The
x and y values used to create the plot are returned invisibly.
If plot and add are FALSE (the default),
the arguments following add abvnonpar or
abvpar, the latter of which plots (or calculates)
the dependence function for ablty
to zero to omit the border.plot.abvnonpar calculates or plots a non-parametric estimate of
the dependence function of the bivariate extreme value distribution.abvpar.
Non-parametric estimates are constructed as follows.
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).Five different estimators of the dependence function can be implemented. They are defined (on $0 \leq w \leq 1$) as follows.
method = "cfg" (Caperaa, Fougeres and Genest, 1997)
$$A_c(w) = \exp\left{ {1-p(w)} \int_{0}^{w}
\frac{H(x) - x}{x(1-x)} \, \mbox{d}x - p(w) \int_{w}^{1}
\frac{H(x) - x}{x(1-x)} \, \mbox{d}x \right}$$
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 = "halltajvidi" (Hall and Tajvidi, 2000)
$$A_h(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{\bar{y}_1
w},\frac{y_{i2}}{\bar{y}_2 (1-w)}\right)\right}^{-1}$$
method = "tdo" (Tiago de Oliveira, 1997)
$$A_t(w) = 1 - \frac{1}{1 + \log n} \sum_{i=1}^n
\min\left(\frac{w}{1+ny_{i1}},\frac{1-w}{1+ny_{i2}}\right)$$
In the estimator $A_h(\cdot)$,
$\bar{y}_j = n^{-1}\sum_{i=1}^n y_{ij}$ for $j = 1,2$.
In the estimator $A_c(\cdot)$, $H(x)$ is the
empirical distribution function of $x_1,\ldots,x_n$, where
$x_i = y_{i1} / (y_{i1} + y_{i2})$ 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 constraint $A_n(0) = A_n(1) = 1$ is satisfied by $A_d(\cdot)$, $A_t(\cdot)$ and $A_h(\cdot)$, and by $A_c(\cdot)$ 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 $$A_n^{'}(w) = \min(1, \max{A_n(w), w, 1-w}).$$ This modification is always implemented.
$A_t(w)$ is the only estimator that is convex.
Convex estimators can be derived from other methods by taking
the convex minorant, which can be achieved by setting convex
to TRUE.
Deheuvels, P. (1991) On the limiting behaviour of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Letters, 12, 429--439.
Hall, P. and Tajvidi, N. (2000) Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli, 6, 835--844.
Pickands, J. (1981) Multivariate extreme value distributions. Proc. 43rd Sess. Int. Statist. Inst., 49, 859--878.
Tiago de Oliveira, J. (1997) Statistical Analysis of Extremes. Pendor.
abvpar, atvnonpar,
fgevbvdata <- rbvevd(100, dep = 0.7, model = "log")
abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE)
abvnonpar(data = bvdata, method = "d", plot = TRUE)
M1 <- fitted(fbvevd(bvdata, model = "log"))
abvpar(dep = M1["dep"], model = "log", plot = TRUE)
abvnonpar(data = bvdata, add = TRUE, lty = 2)Run the code above in your browser using DataLab