Fit the Generalised Pareto Distribution (GPD) to the exceedances (peaks) over a threshold using Maximum Likelihood Estimation (MLE). Optionally, these estimates are plotted as a function of \(k\).
GPDmle(data, start = c(0.1,1), warnings = FALSE, logk = FALSE,
plot = FALSE, add = FALSE, main = "POT estimates of the EVI", ...)POT(data, start = c(0.1,1), warnings = FALSE, logk = FALSE,
plot = FALSE, add = FALSE, main = "POT estimates of the EVI", ...)
A list with following components:
Vector of the values of the tail parameter \(k\).
Vector of the corresponding MLE estimates for the \(\gamma\) parameter of the GPD.
Vector of the corresponding MLE estimates for the \(\sigma\) parameter of the GPD.
Vector of \(n\) observations.
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 \(\sigma\). Default is c(0.1,1).
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 "POT estimates of the EVI".
Additional arguments for the plot function, see plot for more details.
Tom Reynkens based on S-Plus code from Yuri Goegebeur and R code from Klaus Herrmann.
The POT function is the same function but with a different name for compatibility with the old S-Plus code.
For each value of k, we look at the exceedances over the \((k+1)\)th largest observation:
\(X_{n-k+j,n}-X_{n-k,n}\) for \(j=1,...,k\), with \(X_{j,n}\) the \(j\)th largest observation and \(n\) the sample size. The GPD is then fitted to these k exceedances using MLE which yields estimates for the parameters of the GPD: \(\gamma\) and \(\sigma\).
See Section 4.2.2 in Albrecher et al. (2017) for more details.
Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.
Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.
GPDfit, GPDresiduals, EPD
data(soa)
# Look at last 500 observations of SOA data
SOAdata <- sort(soa$size)[length(soa$size)-(0:499)]
# Plot GPD-ML estimates as a function of k
GPDmle(SOAdata, plot=TRUE)
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