ReIns: cProbGPD – R documentation

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cProbGPD

Estimator of small exceedance probabilities and large return periods using censored GPD-MLE

Description

Computes estimates of a small exceedance probability P(X>q) or large return period 1/P(X>q) using the GPD-ML estimator adapted for right censoring.

Usage

cProbGPD(data, censored, gamma1, sigma1, q, plot = FALSE, add = FALSE,
         main = "Estimates of small exceedance probability", ...)

cReturnGPD(data, censored, gamma1, sigma1, 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 `cGPDmle`.
`sigma1`	Vector of n-1 estimates for σ_1 obtained from `cGPDmle`.
`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 `cProbGPD` and `"Estimates of large return period"` for `cReturnGPD`.
`...`	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+ \hat{γ}_1/a_{k,n} \times (q-Z_{n-k,n}))^{-1/\hat{γ}_1}

with Z_{i,n} the i-th order statistic of the data, \hat{γ}_1 the generalised Hill estimator adapted for right censoring and km the Kaplan-Meier estimator for the CDF evaluated in Z_{n-k,n}. The value a is defined as

a_{k,n} = \hat{σ}_1 / \hat{p}_k

with \hat{σ}_1 the ML estimate for σ_1 and \hat{p}_k the proportion of the k largest observations that is non-censored.

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 `cProbGPD`.
`R`	Vector of the corresponding estimates for the return period, only returned for `cReturnGPD`.
`q`	The used large quantile.

Author(s)

Tom Reynkens

References

Einmahl, J.H.J., Fils-Villetard, A. and Guillou, A. (2008). "Statistics of Extremes Under Random Censoring." Bernoulli, 14, 207–227.

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)

# GPD-MLE estimator adapted for right censoring
cpot <- cGPDmle(Z, censored=censored, plot=TRUE)

# Exceedance probability
q <- 10
cProbGPD(Z, gamma1=cpot$gamma1, sigma1=cpot$sigma1,
         censored=censored, q=q, plot=TRUE)
         
# Return period
cReturnGPD(Z, gamma1=cpot$gamma1, sigma1=cpot$sigma1,
         censored=censored, 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