SplineFitDepFct function

Fit a dependence function by spline smoothing

Given estimates for the dependence function of a bivariate extreme value copula at specified points, this function fits a natural cubic smoothing spline, that is constrained to fulfill all the conditions of a dependence function, to the given estimates via quadratic programming.

SplineFitDepFct(x, y = NULL, alpha = 0.01, integ.value)

Arguments

• x, y: vectors giving the coordinates of the points to be approximated. Alternatively a single plotting structure can be specified: see xy.coords.
• alpha: real, the smoothing parameter for the smoothing splines.
• integ.value: real, non-negative value that should be less than two; see Details

Returns

A function, created by splinefun, that evaluates the natural cubic spline that was fitted to the data.

Details

integ.value should be between 0 and 2. If a value is specified, then an additional constraint is added to the quadratic program to ensure that the integeral (over 0 to 1) of the second derivative of the spline is larger or equal to integ.value. Choosing values close to 2 may lead to quadratic programms on which solve.QP reports inconsistent constraints.

Examples

## Data from Hall and Tajvidi (2004, ANZJS)
EstDF2 <- NonparEstDepFct(MaxTemp, convex = FALSE)

## Plot modified Pickands Function and area in which
## dependence function must lie
plot(EstDF2, 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))

## Fit spline to Pickands function and add to plot
splfit <- SplineFitDepFct(EstDF2)
curve(splfit, n = 301, add = TRUE, lty = "dashed")

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

NonparEstDepFct, NewBEVSplineCopula

Author(s)

Nader Tajvidi Nader.Tajvidi@matstat.lu.se

Berwin A Turlach Berwin.Turlach@gmail.com