cProbEPD: Estimator of small exceedance probabilities and large return...

In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

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 \kappa_1 obtained from cEPD.
beta	Vector of n-1 estimates for \beta 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{\gamma}_1, \hat{\kappa}_1 and \hat{\tau}=-\hat{\beta} 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.

See Also

cEPD, ProbEPD, Prob, KaplanMeier

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 documentation built on Nov. 3, 2023, 5:08 p.m.