abvnonpar(x = 0.5, data, nsloc1 = NULL, nsloc2 = NULL,
method = c("cfg", "pickands", "deheuvels", "hall", "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 = "hall"
(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
,
fgev
bvdata <- 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)
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