abvnonpar {evd}

Non-parametric Estimates for Dependence Functions of the Bivariate Extreme Value Distribution

Description

Calculate or plot non-parametric estimates for the dependence function A of the bivariate extreme value distribution.

Usage

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)", ...)

Arguments

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.

epmar

If TRUE, an empirical transformation of the marginals is performed in preference to marginal parametric GEV estimation, and the nsloc arguments are ignored.

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). 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.

k

An integer parameter for the "pot" method. Only the largest k values are used, as described in bvtcplot.

convex

Logical; take the convex minorant?

rev

Logical; reverse the dependence function? This is equivalent to evaluating the function at 1-x.

madj

Performs marginal adjustments for the "pickands" method (see Details).

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 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 abvevd, 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, 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 plot.

Details

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.

Value

abvnonpar calculates or plots a non-parametric estimate of the dependence function of the bivariate extreme value distribution.

Note

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.

References

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.

See Also

abvevd, amvnonpar, bvtcplot, fgev

Examples

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)