Generalized Pareto distribution

The distribution of exceedances over a high threshold. The xi -> 0 limit is the exponential, the mean is sigma/(1 - xi) for xi < 1 and infinite otherwise, and the quantile function inverts in closed form; all three are what the tests check.

Usage

GpdD(sigma, xi, x = NULL, p = NULL)

Arguments

sigma

Positive scale.

xi

Shape.

x

Points at which the CDF and density are evaluated.

p

Probabilities at which quantiles are returned.

Value

List with estimate (mean), cdf, pdf, quantile, mean, variance, upper_endpoint, sigma, xi, n, method.

Details

Formula: F(x) = 1 - (1 + xi x/sigma)^(-1/xi), or 1 - exp(-x/sigma) when xi = 0.

References

Pickands (1975), Statistical inference using extreme order statistics, Annals of Statistics 3(1):119-131. doi:10.1214/aos/1176343003

Examples

GpdD(sigma = 0.5, xi = 5L)
#> $estimate
#> [1] Inf
#> 
#> $cdf
#> [1] 0.3011729 0.3809561 0.4560536 0.5241794
#> 
#> $pdf
#> [1] 0.23294237 0.11255344 0.05180442 0.02321076
#> 
#> $quantile
#> [1]          3.1       9999.9     319999.9  999999999.9
#> 
#> $mean
#> [1] Inf
#> 
#> $variance
#> [1] Inf
#> 
#> $upper_endpoint
#> [1] Inf
#> 
#> $sigma
#> [1] 0.5
#> 
#> $xi
#> [1] 5
#> 
#> $n
#> [1] 4
#> 
#> $method
#> [1] "F(x) = 1 - (1 + xi x/sigma)^(-1/xi), Pickands (1975)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"