P-P plot for testing max stability

The diagnostic, proposed by Gabda, Towe, Wadsworth and Tawn, relies on the fact that, for max-stable vectors on the unit Gumbel scale, the distribution of the maxima is Gumbel distribution with a location parameter equal to the exponent measure. One can thus consider tuples of size m and estimate the location parameter via maximum likelihood and transforming observations to the standard Gumbel scale. Replicates are then pooled and empirical quantiles are defined. The number of combinations of m vectors can be prohibitively large, hence only nmax randomly selected tuples are selected from all possible combinations. The confidence intervals are obtained by a nonparametric bootstrap, by resampling observations with replacement observations for the selected tuples and re-estimating the location parameter. The procedure can be computationally intensive as a result.

Usage

maxstabtest(
  dat,
  m = prod(dim(dat)[-1]),
  nmax = 500L,
  B = 1000L,
  ties.method = "random",
  plot = TRUE
)

Arguments

dat: matrix or array of max-stable observations, typically block maxima. The first dimension should consist of replicates
m: integer indicating how many tuples should be aggregated.
nmax: maximum number of pairs. Default to 500L.
B: number of nonparametric bootstrap replications. Default to 1000L.
ties.method: string indicating the method for rank. Default to "random".
plot: logical indicating whether a graph should be produced (default to TRUE).

Value

a Tukey probability-probability plot with 95

References

Gabda, D.; Towe, R. Wadsworth, J. and J. Tawn, Discussion of ``Statistical Modeling of Spatial Extremes'' by A. C. Davison, S. A. Padoan and M. Ribatet. Statist. Sci. 27 (2012), no. 2, 189--192.

Examples

if (FALSE) {
dat <- mev::rmev(n = 250, d = 100, param = 0.5, model = "log")
maxstabtest(dat, m = 100)
maxstabtest(dat, m = 2, nmax = 100)
dat <- mev::mvrnorm(n = 250, Sigma = diag(0.5, 10) + matrix(0.5, 10, 10), mu = rep(0, 10))
maxstabtest(dat, m = 2, nmax = 100)
maxstabtest(dat, m = ncol(dat))
}