evd (version 2.1-0)

abvnonpar: 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, 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 = "", ...)

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.
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
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
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
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.

Value

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

synopsis

abvnonpar(x = 0.5, data, nsloc1 = NULL, nsloc2 = NULL, method = c("cfg", "pickands", "deheuvels", "hall", "tdo"), convex = FALSE, wf = function(t) t, kmar = NULL, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, xlim = c(0, 1), ylim = c(0.5, 1), xlab = "", ylab = "", ...)

Details

The dependence function $A(\cdot)$ of the bivariate extreme value distribution is defined in abvpar. Non-parametric estimates are constructed as follows. Suppose $(z_{i1},z_{i2})$ for $i=1,\ldots,n$ 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 $$y_{i1} = {1+\hat{s}_1(z_{i1}-\hat{a}_1)/ \hat{b}_1}_{+}^{-1/\hat{s}_1}$$ 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).

Five 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) $$A_c(w) = \exp\left{ {1-p(w)} \int_{0}^{w} \frac{H(x) - x}{x(1-x)} \, \mbox{d}x - p(w) \int_{w}^{1} \frac{H(x) - x}{x(1-x)} \, \mbox{d}x \right}$$ method = "pickands" (Pickands, 1981) $$A_p(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w}, \frac{y_{i2}}{1-w}\right)\right}^{-1}$$

method = "deheuvels" (Deheuvels, 1991) $$A_d(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w}, \frac{y_{i2}}{1-w}\right) - w\sum_{i=1}^n y_{i1} - (1-w) \sum_{i=1}^n y_{i2} + n\right}^{-1}$$

method = "hall" (Hall and Tajvidi, 2000) $$A_h(w) = n\left{\sum_{i=1}^n \min\left(\frac{y_{i1}}{\bar{y}_1 w},\frac{y_{i2}}{\bar{y}_2 (1-w)}\right)\right}^{-1}$$

method = "tdo" (Tiago de Oliveira, 1997) $$A_t(w) = 1 - \frac{1}{1 + \log n} \sum_{i=1}^n \min\left(\frac{w}{1+ny_{i1}},\frac{1-w}{1+ny_{i2}}\right)$$ In the estimator $A_h(\cdot)$, $\bar{y}_j = n^{-1}\sum_{i=1}^n y_{ij}$ for $j = 1,2$. In the estimator $A_c(\cdot)$, $H(x)$ is the empirical distribution function of $x_1,\ldots,x_n$, where $x_i = y_{i1} / (y_{i1} + y_{i2})$ for $i = 1,\ldots,n$, 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 $A_n(\cdot)$ be any estimator of $A(\cdot)$. The constraint $A_n(0) = A_n(1) = 1$ is satisfied by $A_d(\cdot)$, $A_t(\cdot)$ and $A_h(\cdot)$, and by $A_c(\cdot)$ when $p(0) = 0$ and $p(1) = 1$. 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.

$A_t(w)$ 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.

References

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.

See Also

abvpar, atvnonpar, fgev

Examples

Run this code
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)

Run the code above in your browser using DataCamp Workspace