ReIns: GPDmle – R documentation

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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 γ and the second element is the starting value for the estimator of σ. 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 γ should be plotted as a function of k, default is `FALSE`.
`add`	Logical indicating if the estimates of γ 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: γ and σ.

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 γ parameter of the GPD.
`sigma`	Vector of the corresponding MLE estimates for the σ 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.

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

Functions from "Reinsurance: Actuarial and Statistical Aspects"

v1.0.10

GPL (>= 2)

Authors

Tom Reynkens [aut, cre] (<https://orcid.org/0000-0002-5516-5107>), Roel Verbelen [aut] (R code for Mixed Erlang distribution, <https://orcid.org/0000-0002-2347-9240>), Anastasios Bardoutsos [ctb] (Original R code for cEPD estimator), Dries Cornilly [ctb] (Original R code for EVT estimators for truncated data), Yuri Goegebeur [ctb] (Original S-Plus code for basic EVT estimators), Klaus Herrmann [ctb] (Original R code for GPD estimator)

Initial release

2020-05-16