GPDmle function

GPD-ML estimator

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", ...)

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 $j$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.

Returns

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.

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.

Author(s)

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

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)