trMLE: MLE estimator for upper truncated data

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

MLE estimator for upper truncated data

Description

Computes the ML estimator for the extreme value index, adapted for upper truncation, as a function of the tail parameter k (Beirlant et al., 2017). Optionally, these estimates are plotted as a function of k.

Usage

trMLE(data, start = c(1, 1), eps = 10^(-10), 
      plot = TRUE, add = FALSE, main = "Estimates for EVI", ...)

Arguments

data	Vector of n observations.
start	Starting values for \gamma and \tau for the numerical optimisation.
eps	Numerical tolerance, see Details. By default it is equal to 10^(-10).
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 "Estimates of the EVI".
...	Additional arguments for the plot function, see plot for more details.

Details

We compute the MLE for the \gamma and \sigma parameters of the truncated GPD. For numerical reasons, we compute the MLE for \tau=\gamma/\sigma and transform this estimate to \sigma.

The log-likelihood is given by

(k-1) \ln \tau - (k-1) \ln \xi- ( 1 + 1/\xi)\sum_{j=2}^k \ln (1+\tau E_{j,k}) -(k-1) \ln( 1- (1+ \tau E_{1,k})^{-1/\xi})

with E_{j,k} = X_{n-j+1,n}-X_{n-k,n}.

In order to meet the restrictions \sigma=\xi/\tau>0 and 1+\tau E_{j,k}>0 for j=1,\ldots,k, we require the estimates of these quantities to be larger than the numerical tolerance value eps.

See Beirlant 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 estimates for \gamma.
tau	Vector of the corresponding estimates for \tau.
sigma	Vector of the corresponding estimates for \sigma.
conv	Convergence indicator of optim.

Author(s)

Tom Reynkens.

References

Beirlant, J., Fraga Alves, M. I. and Reynkens, T. (2017). "Fitting Tails Affected by Truncation". Electronic Journal of Statistics, 11(1), 2026–2065.

See Also

trDTMLE, trEndpointMLE, trProbMLE, trQuantMLE, trTestMLE, trHill, GPDmle

Examples

# Sample from GPD truncated at 99% quantile
gamma <- 0.5
sigma <- 1.5
X <- rtgpd(n=250, gamma=gamma, sigma=sigma, endpoint=qgpd(0.99, gamma=gamma, sigma=sigma))

# Truncated ML estimator
trmle <- trMLE(X, plot=TRUE, ylim=c(0,2))

TReynkens/ReIns documentation built on Nov. 9, 2023, 1:29 p.m.