GPDmle: GPD-ML estimator

In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

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.

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 jth 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.

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.

Author(s)

Tom Reynkens based on S-Plus code from Yuri Goegebeur and R code from Klaus Herrmann.

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

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)

ReIns documentation built on April 3, 2025, 6:04 p.m.