abvnonpar | R Documentation |
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)", ...)
x |
A vector of values at which the dependence function is
evaluated (ignored if plot or add is |
data |
A matrix or data frame with two columns, which may contain missing values. |
epmar |
If |
nsloc1 , nsloc2 |
A data frame with the same number of rows as
|
method |
The estimation method (see Details). Typically
either |
k |
An integer parameter for the |
convex |
Logical; take the convex minorant? |
rev |
Logical; reverse the dependence function? This is
equivalent to evaluating the function at |
madj |
Performs marginal adjustments for the |
kmar |
In the rare case that the marginal distributions are known, specifies the GEV parameters to be used instead of maximum likelihood estimates. |
plot |
Logical; if |
add |
Logical; add to an existing plot? The existing plot
should have been created using either |
lty , blty |
Function and border line types. Set |
lwd , blwd |
Function and border line widths. |
col |
Line colour. |
xlim , ylim |
x and y-axis limits. |
xlab , ylab |
x and y-axis labels. |
... |
Other high-level graphics parameters to be passed to
|
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
.
abvnonpar
calculates or plots a non-parametric estimate of
the dependence function of the bivariate extreme value distribution.
I have been asked to point out that Hall and Tajvidi (2000) suggest putting a constrained smoothing spline on their modified Pickands estimator, but this is not done here.
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)
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