abvnonpar(x = 0.5, data, epmar = FALSE, nsloc1 = NULL, nsloc2 = NULL,
method = c("cfg", "pickands"), convex = FALSE, rev = FALSE,
madj = 0, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1,
blty = 3, blwd = 1, xlim = c(0, 1), ylim = c(0.5, 1), xlab = "t",
ylab = "A(t)", ...)
TRUE
). $A(1/2)$
is returned by default since it is often a useful summary of
dependence.TRUE
, an empirical transformation of the
marginals is performed in preference to marginal parametric
GEV estimation, and the nsloc
arguments are ignored.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 be give"cfg"
(the default) or "pickands"
.1-x
."pickands"
method (see Details).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
abvevd
, 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.abvevd
.
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.
If epmar
is FALSE
(the default), then
the marginal parameters of the GEV margins 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
).Two 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)
$$\log(A_c(w)) = 1/n \left{ sum_{i=1}^n \log(max[(1-w)y_{i1},
wy_{i1}]) - (1-w)sum_{i=1}^n y_{i1} - w sum_{i=1}^n y_{i2}
\right}$$
method = "pickands"
(Pickands, 1981)
$$A_p(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w},
\frac{y_{i2}}{1-w}\right)\right}^{-1}$$
Two variations on the estimator $A_p(\cdot)$ are
also implemented. If the argument madj = 1
, an adjustment
given in Deheuvels (1991) is applied. If the argument
madj = 2
, an adjustment given in Hall and Tajvidi (2000)
is applied. These are marginal adjustments; they are only
useful when empirical marginal estimation is used.
Let $A_n(\cdot)$ be any estimator of $A(\cdot)$. 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.
Convex estimators can be derived by taking the convex minorant,
which can be achieved by setting convex
to TRUE
.
Pickands, J. (1981) Multivariate extreme value distributions. Proc. 43rd Sess. Int. Statist. Inst., 49, 859--878.
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.
abvevd
, amvnonpar
,
fgev
bvdata <- rbvevd(100, dep = 0.7, model = "log")
abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE)
abvnonpar(data = bvdata, method = "p", plot = TRUE)
M1 <- fitted(fbvevd(bvdata, model = "log"))
abvevd(dep = M1["dep"], model = "log", plot = TRUE)
abvnonpar(data = bvdata, add = TRUE, lty = 2)
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