trMLE: 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", ...)

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.

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.

Author

Tom Reynkens.

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.

References

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

Examples

Run this code

# 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))

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