ReIns: EPD – R documentation

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EPD

EPD estimator

Description

Fit the Extended Pareto Distribution (GPD) to the exceedances (peaks) over a threshold. Optionally, these estimates are plotted as a function of k.

Usage

EPD(data, rho = -1, start = NULL, direct = FALSE, warnings = FALSE, 
    logk = FALSE, plot = FALSE, add = FALSE, main = "EPD estimates of the EVI", ...)

Arguments

`data`	Vector of n observations.
`rho`	A parameter for the ρ-estimator of Fraga Alves et al. (2003) when strictly positive or choice(s) for ρ if negative. Default is `-1`.
`start`	Vector of length 2 containing the starting values for the optimisation. The first element is the starting value for the estimator of γ and the second element is the starting value for the estimator of κ. This argument is only used when `direct=TRUE`. Default is `NULL` meaning the initial value for γ is the Hill estimator and the initial value for κ is 0.
`direct`	Logical indicating if the parameters are obtained by directly maximising the log-likelihood function, see Details. Default is `FALSE`.
`warnings`	Logical indicating if possible warnings from the optimisation function are shown, default is `FALSE`.
`logk`	Logical indicating if the estimates are plotted as a function of \log(k) (`logk=TRUE`) or as a function of k. Default is `FALSE`.
`plot`	Logical indicating if the estimates of γ should be plotted as a function of k, default is `FALSE`.
`add`	Logical indicating if the estimates of γ should be added to an existing plot, default is `FALSE`.
`main`	Title for the plot, default is `"EPD estimates of the EVI"`.
`...`	Additional arguments for the `plot` function, see `plot` for more details.

Details

We fit the Extended Pareto distribution to the relative excesses over a threshold (X/u). The EPD has distribution function F(x) = 1-(x(1+κ-κ x^{τ}))^{-1/γ} with τ = ρ/γ <0<γ and κ>\max(-1,1/τ).

The parameters are determined using MLE and there are two possible approaches: maximise the log-likelihood directly (direct=TRUE) or follow the approach detailed in Beirlant et al. (2009) (direct=FALSE). The latter approach uses the score functions of the log-likelihood.

See Section 4.2.1 of Albrecher et al. (2017) for more details.

Value

A list with following components:

`k`	Vector of the values of the tail parameter k.
`gamma`	Vector of the corresponding estimates for the γ parameter of the EPD.
`kappa`	Vector of the corresponding MLE estimates for the κ parameter of the EPD.
`tau`	Vector of the corresponding estimates for the τ parameter of the EPD using Hill estimates and values for ρ.

Author(s)

Tom Reynkens

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant, J., Joossens, E. and Segers, J. (2009). "Second-Order Refined Peaks-Over-Threshold Modelling for Heavy-Tailed Distributions." Journal of Statistical Planning and Inference, 139, 2800–2815.

Fraga Alves, M.I. , Gomes, M.I. and de Haan, L. (2003). "A New Class of Semi-parametric Estimators of the Second Order Parameter." Portugaliae Mathematica, 60, 193–214.

Examples

data(secura)

# EPD estimates for the EVI
epd <- EPD(secura$size, plot=TRUE)

# Compute return periods
ReturnEPD(secura$size, 10^10, gamma=epd$gamma, kappa=epd$kappa, 
          tau=epd$tau, plot=TRUE)

ReIns

Functions from "Reinsurance: Actuarial and Statistical Aspects"

v1.0.10

GPL (>= 2)

Authors

Tom Reynkens [aut, cre] (<https://orcid.org/0000-0002-5516-5107>), Roel Verbelen [aut] (R code for Mixed Erlang distribution, <https://orcid.org/0000-0002-2347-9240>), Anastasios Bardoutsos [ctb] (Original R code for cEPD estimator), Dries Cornilly [ctb] (Original R code for EVT estimators for truncated data), Yuri Goegebeur [ctb] (Original S-Plus code for basic EVT estimators), Klaus Herrmann [ctb] (Original R code for GPD estimator)

Initial release

2020-05-16

EPD