trMLE function

MLE estimator for upper truncated data

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 $k$.

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

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.

Returns

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.

References

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

Author(s)

Tom Reynkens.

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))