ReIns (version 1.0.7)

trDT: 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 \(n\) observations.

r

Trimming parameter, default is 1 (no trimming).

gamma

Vector of \(n-1\) estimates for the EVI obtained from trHill.

plot

Logical indicating if the estimates of \(D_T\) should be plotted as a function of \(k\), default is FALSE.

add

Logical indicating if the estimates of \(D_T\) 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.

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

DT

Vector of the corresponding estimates for the truncation odds \(D_T\).

Details

The truncation odds is defined as $$D_T=(1-F(T))/F(T)$$ with \(T\) the upper truncation point and \(F\) the CDF of the untruncated distribution (e.g. Pareto distribution).

We estimate this truncation odds as $$\hat{D}_T=\max\{ (k+1)/(n+1) ( R_{r,k,n}^{1/\gamma_k} - 1/(k+1) ) / (1-R_{r,k,n}^{1/\gamma_k}), 0\}$$ with \(R_{r,k,n} = X_{n-k,n} / X_{n-r+1,n}\).

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

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

# NOT RUN {
# 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))
# }