trDT {ReIns}R Documentation

Truncation odds

Description

Estimates of truncation odds of the truncated probability mass under the untruncated distribution using truncated Hill.

Usage

trDT(data, r = 1, gamma, plot = FALSE, add = FALSE, main = "Estimates of DT", ...)

Arguments

data

Vector of nn observations.

r

Trimming parameter, default is 1 (no trimming).

gamma

Vector of n1n-1 estimates for the EVI obtained from trHill.

plot

Logical indicating if the estimates of DTD_T should be plotted as a function of kk, default is FALSE.

add

Logical indicating if the estimates of DTD_T should be added to an existing plot, default is FALSE.

main

Title for the plot, default is "Estimates of DT".

...

Additional arguments for the plot function, see plot for more details.

Details

The truncation odds is defined as

DT=(1F(T))/F(T)D_T=(1-F(T))/F(T)

with TT the upper truncation point and FF the CDF of the untruncated distribution (e.g. Pareto distribution).

We estimate this truncation odds as

D^T=max{(k+1)/(n+1)(Rr,k,n1/γk1/(k+1))/(1Rr,k,n1/γk),0}\hat{D}_T=\max\{ (k+1)/(n+1) ( R_{r,k,n}^{1/\gamma_k} - 1/(k+1) ) / (1-R_{r,k,n}^{1/\gamma_k}), 0\}

with Rr,k,n=Xnk,n/Xnr+1,nR_{r,k,n} = X_{n-k,n} / X_{n-r+1,n}.

See Beirlant et al. (2016) or Section 4.2.3 of Albrecher et al. (2017) for more details.

Value

A list with following components:

k

Vector of the values of the tail parameter kk.

DT

Vector of the corresponding estimates for the truncation odds DTD_T.

Author(s)

Tom Reynkens based on R code of Dries Cornilly.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant, J., Fraga Alves, M.I. and Gomes, M.I. (2016). "Tail fitting for Truncated and Non-truncated Pareto-type Distributions." Extremes, 19, 429–462.

See Also

trHill, trEndpoint, trQuant, trDTMLE

Examples

# Sample from truncated Pareto distribution.
# truncated at 99% quantile
shape <- 2
X <- rtpareto(n=1000, shape=shape, endpoint=qpareto(0.99, shape=shape))

# Truncated Hill estimator
trh <- trHill(X, plot=TRUE, ylim=c(0,2))

# Truncation odds
dt <- trDT(X, gamma=trh$gamma, plot=TRUE, ylim=c(0,0.05))

[Package ReIns version 1.0.14 Index]