Calculate or plot non-parametric estimates for the dependence function \(A\) of the bivariate extreme value distribution.
abvnonpar(x = 0.5, data, epmar = FALSE, nsloc1 = NULL,
nsloc2 = NULL, method = c("cfg", "pickands", "tdo", "pot"),
k = nrow(data)/4, convex = FALSE, rev = FALSE, madj = 0,
kmar = NULL, 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)", ...)
abvnonpar
calculates or plots a non-parametric estimate of
the dependence function of the bivariate extreme value distribution.
A vector of values at which the dependence function is
evaluated (ignored if plot or add is TRUE
). \(A(1/2)\)
is returned by default since it is often a useful summary of
dependence.
A matrix or data frame with two columns, which may contain missing values.
If TRUE
, an empirical transformation of the
marginals is performed in preference to marginal parametric
GEV estimation, and the nsloc
arguments are ignored.
A data frame with the same number of rows as
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 given
as an alternative to a single column data frame.
The estimation method (see Details). Typically
either "cfg"
(the default) or "pickands"
. The method
"tdo"
performs poorly and is not recommended. The method
"pot"
is for peaks over threshold modelling where only
large data values are used for estimation.
An integer parameter for the "pot"
method. Only the
largest k
values are used, as described in
bvtcplot
.
Logical; take the convex minorant?
Logical; reverse the dependence function? This is
equivalent to evaluating the function at 1-x
.
Performs marginal adjustments for the "pickands"
method (see Details).
In the rare case that the marginal distributions are known, specifies the GEV parameters to be used instead of maximum likelihood estimates.
Logical; if 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
are ignored.
Logical; add to an existing plot? The existing plot
should have been created using either abvnonpar
or
abvevd
, the latter of which plots (or calculates)
the dependence function for a number of parametric models.
Function and border line types. Set blty
to zero to omit the border.
Function and border line widths.
Line colour.
x and y-axis limits.
x and y-axis labels.
Other high-level graphics parameters to be passed to
plot
.
The dependence function \(A(\cdot)\) of the bivariate
extreme value distribution is defined in 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)
$$\log(A_c(w)) = \frac{1}{n} \left\{ \sum_{i=1}^n \log(\max[(1-w)y_{i1},
wy_{i1}]) - (1-w)\sum_{i=1}^n y_{i1} - w \sum_{i=1}^n y_{i2}
\right\}$$
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
.
Caperaa, P. Fougeres, A.-L. and Genest, C. (1997) A non-parametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567--577.
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
,
bvtcplot
, fgev
bvdata <- rbvevd(100, dep = 0.7, model = "log")
abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE)
abvnonpar(data = bvdata, method = "pick", plot = TRUE)
M1 <- fitted(fbvevd(bvdata, model = "log"))
abvevd(dep = M1["dep"], model = "log", plot = TRUE)
abvnonpar(data = bvdata, add = TRUE, lty = 2)
Run the code above in your browser using DataLab