# NonparEstDepFct: Nonparametric estimator of bivariate dependence function In SimCop: Simulate from Arbitrary Copulae

## Description

Function to calculate nonparametric estimates of the dependence functions of bivariate extreme value copula.

## Usage

 ```1 2``` ```NonparEstDepFct(x, y = NULL, w.length = 101, transf.to.frechet = TRUE, convex.hull = TRUE, verbose = FALSE) ```

## Arguments

 `x, y` vectors giving the observations of the extreme values. Alternatively a single plotting structure can be specified: see `xy.coords`. `w.length` number of grid points (using an equidistant grid from 0 to 1) on which the dependence function is estimated. `transf.to.frechet` logical, controls whether `x` and `y` are first transformed to have standard Fréchet margins: see ‘Details’; defaults to `TRUE`. `convex.hull` logical, controls whether the convex hull of the modified Pickands estimator is returned; defaults to `TRUE`. `verbose` logical, controls whether progress messages are given; defaults to `FALSE`.

## Details

If `transf.to.frechet` is `TRUE`, the default, then a generalised extreme value (GEV) distribution is fitted to each margin and the fitted parameters are used to transform the data to have standard Fréchet margins. The parameterisation of the cumulative distribution of the GEV that is used is, if γ <> 0:

G(z) = exp( -{1 + γ [(z - μ)/σ] }^(-1/γ) )

and for γ = 0:

G(z) = exp( -exp(-z) )

If γ < 0, then the support of the GEV is the interval (-∞, μ - σ/γ], while it is [μ - σ/γ, ∞) if γ > 0. For γ = 0, the support is the real line.

If `verbose` is `TRUE`, not the default, and `transf.to.frechet` is `TRUE`, the estimates for the fitted GEV distribution are printed out using `cat`.

## Value

A list with two named components. The component `x` contains a vector with the grid points at which the dependence function was estimated. The component `y` contains the estimated dependence functions.

## References

Hall, P. and Tajvidi, N. (2000). Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli 6(5), 835–844. Doi:10.2307/3318758.

Hall, P. and Tajvidi, N. (2004). Prediction regions for bivariate extreme events. Australian & New Zealand Journal of Statistics 46(1), 99–112. Doi:10.1111/j.1467-842X.2004.00316.x.

`SplineFitDepFct`
 ```1 2 3 4 5 6 7``` ```## Data from Hall and Tajvidi (2004, ANZJS) EstDF1 <- NonparEstDepFct(MaxTemp) ## Plot modified Pickands Function and area in which ## dependence function must lie plot(EstDF1, ylim = c(0.5,1), xlab = "w", ylab = "A(w)", type="l", lty="longdash") polygon(c(0, 0.5, 1, 0), c(1, 0.5, 1, 1)) ```