EPD function

EPD estimator

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

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 $\rho$-estimator of Fraga Alves et al. (2003) when strictly positive or choice(s) for $\rho$ 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 $\gamma$ and the second element is the starting value for the estimator of $\kappa$. This argument is only used when direct=TRUE. Default is NULL meaning the initial value for $\gamma$ is the Hill estimator and the initial value for $\kappa$ 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 $\gamma$ should be plotted as a function of $k$, default is FALSE.
• add: Logical indicating if the estimates of $\gamma$ 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 $F(x) = 1-(x(1+\kappa-\kappa x^{\tau}))^{-1/\gamma}$

with $\tau = \rho/\gamma <0<\gamma$ and $\kappa>\max(-1,1/\tau)$.

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.

Returns

A list with following components: - k: Vector of the values of the tail parameter $k$.

• gamma: Vector of the corresponding estimates for the $\gamma$ parameter of the EPD.

• kappa: Vector of the corresponding MLE estimates for the $\kappa$ parameter of the EPD.

• tau: Vector of the corresponding estimates for the $\tau$ parameter of the EPD using Hill estimates and values for $\rho$.

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.

Author(s)

Tom Reynkens

See Also

GPDmle, ProbEPD

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)