ReIns: cQuantGH – R documentation

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cQuantGH

Estimator of large quantiles using censored Hill

Description

Computes estimates of large quantiles Q(1-p) using the estimates for the EVI obtained from the generalised Hill estimator adapted for right censoring.

Usage

cQuantGH(data, censored, gamma1, p, plot = FALSE, add = FALSE, 
         main = "Estimates of extreme quantile", ...)

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 `cgenHill`.
`p`	The exceedance probability of the quantile (we estimate Q(1-p) for p small).
`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 extreme quantile"`.
`...`	Additional arguments for the `plot` function, see `plot` for more details.

Details

The quantile is estimated as

\hat{Q}(1-p)= Z_{n-k,n} + a_{k,n} ( ( (1-km)/p)^{\hat{γ}_1} -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} = Z_{n-k,n} H_{k,n} (1-S_{Z,k,n}) / \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, and

S_{Z,k,n} = 1 - (1-M_1^2/M_2)^(-1) / 2

with

M_l = =1/k∑_{j=1}^k (\log X_{n-j+1,n}- \log X_{n-k,n})^l.

Value

A list with following components:

`k`	Vector of the values of the tail parameter k.
`Q`	Vector of the corresponding quantile estimates.
`p`	The used exceedance probability.

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)

# Generalised Hill estimator adapted for right censoring
cghill <- cgenHill(Z, censored=censored, plot=TRUE)

# Large quantile
p <- 10^(-4)
cQuantGH(Z, gamma1=cghill$gamma, censored=censored, p=p, 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