adf_gof {ReturnCurves}

Goodness of fit of the Angular Dependence function estimates

Description

Assessment of the goodness of fit of the angular dependence function estimates \(\lambda(\omega)\) following the procedure of Murphy-Barltrop et al. (2024).

Usage

adf_gof(adf, ray, blocksize = 1, nboot = 250, alpha = 0.05)

Arguments

adf

An S4 object of class adf_est.class. See adf_est for more details.

ray

Ray \(\omega\) to be considered on the goodness of fit assessment.

blocksize

Size of the blocks for the block bootstrap procedure. If 1 (default), then a standard bootstrap approach is applied.

nboot

Number of bootstrap samples to be taken. Default is 250 samples.

alpha

Significance level to compute the \((1-\alpha)\)% tolerance intervals. Default is 0.05.

Details

Define the min-projection variable as \(t^1_\omega = t_\omega - u_\omega | t_\omega > u_\omega\), then variable \(\lambda(\omega)T^1_\omega \sim Exp(1)\) as \(u_\omega \to \infty\) for all \(\omega \in [0,1]\).

Let \(F^{-1}_E\) denote the inverse of the cumulative distribution function of a standard exponential variable and \(T^1_{(i)}\) denote the \(i\)-th ordered increasing statistic, \(i = 1, \ldots, n\). Function plot shows a QQ plot between the model and empirical exponential quantiles, i.e. points \(\left(F^{-1}_E\left(\frac{i}{n+1}\right), T^1_{(i)}\right)\), along with the line \(y=x\). Uncertainty is obtained via a (block) bootstrap procedure and shown by the grey region on the plot. A good fit is shown by agreement of model and empirical quantiles, i.e. points should lie close to the line \(y=x\). In addition, line \(y = x\) should mainly lie within the \((1-\alpha)\)% tolerance intervals.

We note that, if the grid for \(\omega\) used to estimate the Angular Dependence Function (ADF) does not contain ray, then the closest \(\omega\) in the grid is used to assess the goodness-of-fit of the ADF.

Value

An object of S4 class adf_gof.class. This object returns the arguments of the function and an extra slot gof which is a list containing:

model

A vector containing the model quantiles.

empirical

A vector containing the empirical quantiles.

lower

A vector containing the lower bound of the tolerance interval.

upper

A vector containing the upper bound of the tolerance interval.

Note

It is recommended to assess the goodness-of-fit of \(\lambda(\omega)\) for a few values of \(\omega\).

References

Murphy-Barltrop CJR, Wadsworth JL, Eastoe EF (2024). “Improving estimation for asymptotically independent bivariate extremes via global estimators for the angular dependence function.” 2303.13237.

Examples

library(ReturnCurves)

data(airdata)

n <- dim(airdata)[1]

margdata <- margtransf(airdata)

lambda <- adf_est(margdata = margdata, method = "hill")

# blocksize to account for temporal dependence
gof <- adf_gof(adf = lambda, ray = 0.4, blocksize = 10)

plot(gof)

# To see the the S4 object's slots
str(gof)

# To access the list of vectors
gof@gof