SimCop (version 0.7.0)

NonparEstDepFct: Nonparametric estimator of bivariate dependence function

Description

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

Usage

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

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.

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.

Author

Nader Tajvidi <[e-mail]>

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 \(\gamma \neq 0\): $$G(z) = \exp\left(-\left[1+\gamma\left(\frac{z-\mu}{\sigma}\right)\right]^{-1/\gamma}\right)$$ and for \(\gamma = 0\): $$G(z) = \exp(-\exp(-z))$$ If \(\gamma < 0\), then the support of the GEV is the interval \((-\infty, \mu - \sigma/\gamma]\), while it is \([\mu - \sigma/\gamma, \infty)\) if \(\gamma > 0\). For \(\gamma = 0\), 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.

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.

See Also

SplineFitDepFct

Examples

## 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))