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", ...)

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 $D_T$.

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.

Author

Tom Reynkens based on R code of Dries Cornilly.

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.

Examples

Run this code

# 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))

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