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", ...)Vector of \(n\) observations.
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.
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.
Logical indicating if the parameters are obtained by directly maximising the log-likelihood function, see Details. Default is FALSE.
Logical indicating if possible warnings from the optimisation function are shown, default is FALSE.
Logical indicating if the estimates are plotted as a function of \(\log(k)\) (logk=TRUE) or as a function of \(k\). Default is FALSE.
Logical indicating if the estimates of \(\gamma\) should be plotted as a function of \(k\), default is FALSE.
Logical indicating if the estimates of \(\gamma\) should be added to an existing plot, default is FALSE.
Title for the plot, default is "EPD estimates of the EVI".
Additional arguments for the plot function, see plot for more details.
A list with following components:
Vector of the values of the tail parameter \(k\).
Vector of the corresponding estimates for the \(\gamma\) parameter of the EPD.
Vector of the corresponding MLE estimates for the \(\kappa\) parameter of the EPD.
Vector of the corresponding estimates for the \(\tau\) parameter of the EPD using Hill estimates and values for \(\rho\).
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.
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.
# 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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