# trMLE: MLE estimator for upper truncated data In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

## 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

 1 2 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 γ and τ for the numerical optimisation. eps Numerical tolerance, see Details. By default it is equal to 10^(-10). 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 "Estimates of the EVI". ... Additional arguments for the plot function, see plot for more details.

## Details

We compute the MLE for the γ and σ parameters of the truncated GPD. For numerical reasons, we compute the MLE for τ=γ/σ and transform this estimate to σ.

The log-likelihood is given by

(k-1) \ln τ - (k-1) \ln ξ- ( 1 + 1/ξ)∑_{j=2}^k \ln (1+τ E_{j,k}) -(k-1) \ln( 1- (1+ τ E_{1,k})^{-1/ξ})

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

In order to meet the restrictions σ=ξ/τ>0 and 1+τ E_{j,k}>0 for j=1,…,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 γ. tau Vector of the corresponding estimates for τ. sigma Vector of the corresponding estimates for σ. conv Convergence indicator of optim.

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.

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

## Examples

 1 2 3 4 5 6 7 # 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)) 

### Example output




ReIns documentation built on July 2, 2020, 4:03 a.m.