trDT: Truncation odds

In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

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.

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.

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.

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))

ReIns documentation built on Nov. 3, 2023, 5:08 p.m.