Peaks-over-threshold GPD fit + scale-invariance check

Formula: fit GPD on x[x>u]; rate zeta_u = N_u/n

Usage

Evpot(x, u)

Arguments

x

Sample.

u

Threshold.

Value

List with sigma, xi, zeta_u, estimate, n_exceed, n, nll, modified_scale, method.

Details

Maximum likelihood in Grimshaw's one-parameter reduction: with t = xi/sigma the profile has xi(t) = mean(log(1 + t y)) and sigma(t) = xi(t)/t, leaving a single bounded search. The exponential (xi = 0) fit, sigma = mean(y), is evaluated separately and kept if its likelihood is higher. The modified scale sigma - xi u is reported: it is invariant to the threshold when the GPD model holds.

References

Davison & Smith (1990), JRSS B 52(3):393-442; Grimshaw (1993), Technometrics 35(2):185-191.

Examples

set.seed(1)
x <- rexp(200)
Evpot(x, u = 1)
#> $sigma
#> [1] 0.8572761
#> 
#> $xi
#> [1] -0.05491528
#> 
#> $zeta_u
#> [1] 0.4
#> 
#> $estimate
#> [1] -0.05491528
#> 
#> $n_exceed
#> [1] 80
#> 
#> $n
#> [1] 200
#> 
#> $nll
#> [1] 63.28716
#> 
#> $modified_scale
#> [1] 0.9121914
#> 
#> $method
#> [1] "GPD maximum likelihood on threshold exceedances"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"