ReIns: trProb – R documentation

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trProb

Estimator of small exceedance probabilities using truncated Hill

Description

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

Usage

trProb(data, r = 1, gamma, q, warnings = TRUE, plot = FALSE, add = FALSE, 
       main = "Estimates of small exceedance probability", ...)

Arguments

`data`	Vector of n observations.
`r`	Trimming parameter, default is `1` (no trimming).
`gamma`	Vector of n-1 estimates for the EVI obtained from `trHill`.
`q`	The used large quantile (we estimate P(X>q) for q large).
`warnings`	Logical indicating if warnings are shown, default is `TRUE`.
`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}(X>q)=(k+1)/(n+1) ( (q/X_{n-k,n})^{-1/γ_k} - R_{r,k,n}^{1/\hat{γ}_k} ) / (1- R_{r,k,n}^{1/\hat{γ}_k})

with R_{r,k,n} = X_{n-k,n} / X_{n-r+1,n} and \hat{γ}_k the Hill estimates adapted for truncation.

See Beirlant et al. (2016) or Section 4.2.3 of Albrecher 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 based on R code of Dries Cornilly.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant, J., Fraga Alves, M.I. and Gomes, M.I. (2016). "Tail fitting for Truncated and Non-truncated Pareto-type Distributions." Extremes, 19, 429–462.

Examples

# Sample from truncated Pareto distribution.
# truncated at 99% quantile
shape <- 2
X <- rtpareto(n=1000, shape=shape, endpoint=qpareto(0.99, shape=shape))

# Truncated Hill estimator
trh <- trHill(X, plot=TRUE, ylim=c(0,2))

# Small probability
trProb(X, gamma=trh$gamma, q=8, plot=TRUE)

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