ReIns: cProbEPD – R documentation

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cProbEPD

Estimator of small exceedance probabilities and large return periods using censored EPD

Description

Computes estimates of a small exceedance probability P(X>q) or large return period 1/P(X>q) using the parameters from the EPD fit adapted for right censoring.

Usage

cProbEPD(data, censored, gamma1, kappa1, beta, q, plot = FALSE, add = FALSE,
         main = "Estimates of small exceedance probability", ...)

cReturnEPD(data, censored, gamma1, kappa1, beta, q, plot = FALSE, add = FALSE,
           main = "Estimates of large return period", ...)

Arguments

`data`	Vector of n observations.
`censored`	A logical vector of length n indicating if an observation is censored.
`gamma1`	Vector of n-1 estimates for the EVI obtained from `cEPD`.
`kappa1`	Vector of n-1 estimates for κ_1 obtained from `cEPD`.
`beta`	Vector of n-1 estimates for β obtained from `cEPD`.
`q`	The used large quantile (we estimate P(X>q) or 1/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"` for `cProbEPD` and `"Estimates of large return period"` for `cReturnEPD`.
`...`	Additional arguments for the `plot` function, see `plot` for more details.

Details

The probability is estimated as

\hat{P}(X>q)= (1-km) \times (1-F(q))

with F the CDF of the EPD with estimated parameters \hat{γ}_1, \hat{κ}_1 and \hat{τ}=-\hat{β} and km the Kaplan-Meier estimator for the CDF evaluated in Z_{n-k,n} (the (k+1)-th largest data point).

Value

A list with following components:

`k`	Vector of the values of the tail parameter k.
`P`	Vector of the corresponding probability estimates, only returned for `cProbEPD`.
`R`	Vector of the corresponding estimates for the return period, only returned for `cReturnEPD`.
`q`	The used large quantile.

Author(s)

Tom Reynkens.

References

Beirlant, J., Bardoutsos, A., de Wet, T. and Gijbels, I. (2016). "Bias Reduced Tail Estimation for Censored Pareto Type Distributions." Statistics & Probability Letters, 109, 78–88.

Examples

# Set seed
set.seed(29072016)

# Pareto random sample
X <- rpareto(500, shape=2)

# Censoring variable
Y <- rpareto(500, shape=1)

# Observed sample
Z <- pmin(X, Y)

# Censoring indicator
censored <- (X>Y)

# EPD estimator adapted for right censoring
cepd <- cEPD(Z, censored=censored, plot=TRUE)

# Small exceedance probability
q <- 10
cProbEPD(Z, censored=censored, gamma1=cepd$gamma1,
        kappa1=cepd$kappa1, beta=cepd$beta, q=q, plot=TRUE)

# Return period
cReturnEPD(Z, censored=censored, gamma1=cepd$gamma1,
        kappa1=cepd$kappa1, beta=cepd$beta, q=q, 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