abvnonpar {evd} | R Documentation |
Calculate or plot non-parametric estimates for the dependence function A of the bivariate extreme value distribution.
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 = "", ...)
x |
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. |
data |
A matrix or data frame with two columns, which may contain missing values. |
nsloc1, nsloc2 |
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. |
method |
The estimation method (see Details). The
"cfg" method is used by default. |
convex |
Logical; take the convex minorant? |
wf |
The weight function used in the "cfg"
method (see Details). The function must be vectorized. |
plot |
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. |
add |
Logical; add to an existing plot? The existing plot
should have been created using either abvnonpar or
abvpar , the latter of which plots (or calculates)
the dependence function for a number of parametric models. |
lty, blty |
Function and border line types. Set blty
to zero to omit the border. |
lwd |
Line width. |
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
plot . |
The dependence function A() of the bivariate
extreme value distribution is defined in abvpar
.
Non-parametric estimates are constructed as follows.
Suppose (z_{i1},z_{i2}) for i=1,...,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 + s'_1(z_{i1}-a'_1)/b'_1}^(-1/s'_1)
and
y_{i2} = {1 + s'_2(z_{i2}-a'_2)/b'_2}^(-1/s'_2)
for i = 1,...,n, where
(a'_1,b'_1,s'_1) and
(a'_2,b'_2,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 <= w <= 1) as follows.
method = "cfg"
(Caperaa, Fougeres and Genest, 1997)
A_c(w) = exp{ [1-p(w)] integral_0^w (H(x) - x)/[x(1-x)] dx - p(w) integral_w^1 (H(x) - x)/[x(1-x)] dx }
method = "pickands"
(Pickands, 1981)
A_p(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}/(1-w)]}
method = "deheuvels"
(Deheuvels, 1991)
A_d(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}(1-w)] - w sum_{i=1}^n y_{i1} - (1-w) sum_{i=1}^n y_{i2} + n}
method = "hall"
(Hall and Tajvidi, 2000)
A_h(w) = n (sum_{i=1}^n min[y_{i1}/(by_1 w), y_{i2}/(by_2 (1-w))])^{-1}
method = "tdo"
(Tiago de Oliveira, 1997)
A_t(w) = 1 - 1/(1 + log n) sum_{i=1}^n min[w/(1 + n y_{i1}), (1 - w)/(1 + n y_{i2})]
In the estimator A_h(),
by_j = (sum_{i=1}^n y_{ij})/n for j = 1,2.
In the estimator A_c(), H(x) is the
empirical distribution function of x_1,...,x_n, where
x_i = y_{i1} / (y_{i1} + y_{i2}) for i = 1,...,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() be any estimator of A(). The constraint A_n(0) = A_n(1) = 1 is satisfied by A_d(), A_t() and A_h(), and by A_c() when p(0) = 0 and p(1) = 1. None of the estimators satisfy max(w,1-w) <= A_n(w) <= 1 for all 0 <= w <= 1. 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
.
abvnonpar
calculates or plots a non-parametric estimate of
the dependence function of the bivariate extreme value distribution.
Appendix A of the User's Guide contains a short simulation study that compares the estimators defined above. The estimators A_p(), A_d() and A_h() are very similar, and may not be distinguishable when plotted.
Caperaa, P. Fougeres, A.-L. and Genest, C. (1997) A non-parametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.
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.
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)