ReIns: trProbMLE – R documentation

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trProbMLE

Estimator of small exceedance probabilities using truncated MLE

Description

Computes estimates of a small exceedance probability P(X>q) using the estimates for the EVI obtained from the ML estimator adapted for upper truncation.

Usage

trProbMLE(data, gamma, tau, DT, q, plot = FALSE, add = FALSE, 
          main = "Estimates of small exceedance probability", ...)

Arguments

`data`	Vector of n observations.
`gamma`	Vector of n-1 estimates for the EVI obtained from `trMLE`.
`tau`	Vector of n-1 estimates for the τ obtained from `trMLE`.
`DT`	Vector of n-1 estimates for the truncation odds obtained from `trDTMLE`.
`q`	The used large quantile (we estimate P(X>q) for q large).
`plot`	Logical indicating if the estimates should be plotted as a function of k, default is `FALSE`.
`add`	Logical indicating if the estimates should be added to an existing plot, default is `FALSE`.
`main`	Title for the plot, default is `"Estimates of small exceedance probability"`.
`...`	Additional arguments for the `plot` function, see `plot` for more details.

Details

The probability is estimated as

\hat{p}_{T,k}(q) = (1+ \hat{D}_{T,k}) (k+1)/(n+1) (1+\hatτ _k(q-X_{n-k,n}))^{-1/\hat{ξ}_k} -\hat{D}_{T,k}

with \hat{γ}_k and \hat{τ}_k the ML estimates adapted for truncation and \hat{D}_T the estimates for the truncation odds.

See Beirlant et al. (2017) for more details.

Value

A list with following components:

`k`	Vector of the values of the tail parameter k.
`P`	Vector of the corresponding probability estimates.
`q`	The used large quantile.

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))

# Truncation odds
dtmle <- trDTMLE(X, gamma=trmle$gamma, tau=trmle$tau, plot=FALSE)

# Small exceedance probability
trProbMLE(X, gamma=trmle$gamma, tau=trmle$tau, DT=dtmle$DT, plot=TRUE, q=26, ylim=c(0,0.005))

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