Threshold selection diagnostics for peaks-over-threshold analysis

The two standard plots for choosing a peaks-over-threshold cutoff. The mean residual life plot is the sample mean of excesses above each candidate threshold, which is linear in the threshold once a generalized Pareto model holds. The parameter stability plot fits a generalized Pareto distribution at each candidate threshold; the modified scale and the shape are approximately constant above the true threshold.

Usage

threshold_diagnostics(
  x,
  u_min,
  u_max,
  n_thresholds = 20,
  confidence_level = 0.95,
  method = c("mean_residual_life", "parameter_stability")
)

Arguments

x: numeric vector of observations.
u_min, u_max: the range of candidate thresholds to scan.
n_thresholds: number of equally spaced candidate thresholds in [u_min, u_max].
confidence_level: confidence level for the interval bands.
method: "mean_residual_life" (the default) or "parameter_stability".

Value

A named list of parallel vectors. Both methods return threshold and exceedance_count; "mean_residual_life" adds mean_excess, lower_ci, and upper_ci, and "parameter_stability" adds modified_scale, shape, and their confidence bounds. The columns the chosen method does not populate come back empty.

Details

Candidate thresholds with too few exceedances are dropped (fewer than 5 for mean residual life, fewer than 10 for parameter stability), as are thresholds where the fit fails, so the returned vectors are usually shorter than n_thresholds.

References

Coles (2001), An Introduction to Statistical Modeling of Extreme Values, Section 4.3; Davison and Smith (1990).

Examples

d <- distribution("GeneralizedPareto", c(0, 100, 0.1))
x <- dist_random(d, 500, seed = 42)
mrl <- threshold_diagnostics(x, u_min = 0, u_max = 200, n_thresholds = 10)
data.frame(threshold = mrl$threshold, mean_excess = mrl$mean_excess)
#>    threshold mean_excess
#> 1    0.00000    92.23722
#> 2   22.22222    89.63642
#> 3   44.44444    89.90720
#> 4   66.66667    86.18157
#> 5   88.88889    85.77808
#> 6  111.11111    87.18757
#> 7  133.33333    87.71434
#> 8  155.55556    83.13719
#> 9  177.77778    77.51777
#> 10 200.00000    72.51115