ReIns (version 1.0.7)

GPDmle: GPD-ML estimator

Description

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 \(k\).

Usage

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", ...)

Arguments

data

Vector of \(n\) observations.

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 \(\sigma\). Default is c(0.1,1).

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 "POT estimates of the EVI".

…

Additional arguments for the plot function, see plot for more details.

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

gamma

Vector of the corresponding MLE estimates for the \(\gamma\) parameter of the GPD.

sigma

Vector of the corresponding MLE estimates for the \(\sigma\) parameter of the GPD.

Details

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.

References

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.

See Also

GPDfit, GPDresiduals, EPD

Examples

# NOT RUN {
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)
# }