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.

Value

A list with following components:

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 $ρ$ .

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.

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

Run this code

# NOT RUN {
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)
# }

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