ReIns: trMLE – R documentation

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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", ...)

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`.

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.

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))

ReIns

Functions from "Reinsurance: Actuarial and Statistical Aspects"

v1.0.10

GPL (>= 2)

Authors

Tom Reynkens [aut, cre] (<https://orcid.org/0000-0002-5516-5107>), Roel Verbelen [aut] (R code for Mixed Erlang distribution, <https://orcid.org/0000-0002-2347-9240>), Anastasios Bardoutsos [ctb] (Original R code for cEPD estimator), Dries Cornilly [ctb] (Original R code for EVT estimators for truncated data), Yuri Goegebeur [ctb] (Original S-Plus code for basic EVT estimators), Klaus Herrmann [ctb] (Original R code for GPD estimator)

Initial release

2020-05-16