ReIns: cProbMOM – R documentation

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cProbMOM

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

Description

Computes estimates of a small exceedance probability P(X>q) or large return period 1/P(X>q) using the Method of Moments estimates for the EVI adapted for right censoring.

Usage

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

cReturnMOM(data, censored, gamma1, 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 `cMoment`.
`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 `cProbMOM` and `"Estimates of large return period"` for `cReturnMOM`.
`...`	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 MOM 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}= Z_{n-k,n} H_{k,n} (1-\min(\hat{γ}_1,0)) /\hat{p}_k

with H_{k,n} the ordinary Hill estimator 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 `cProbMOM`.
`R`	Vector of the corresponding estimates for the return period, only returned for `cReturnMOM`.
`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)

# Moment estimator adapted for right censoring
cmom <- cMoment(Z, censored=censored, plot=TRUE)

# Small exceedance probability
q <- 10
cProbMOM(Z, censored=censored, gamma1=cmom$gamma1, q=q, plot=TRUE)

# Return period
cReturnMOM(Z, censored=censored, gamma1=cmom$gamma1, 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