trDTMLE: Truncation odds

In TReynkens/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 MLE.

Usage

trDTMLE(data, gamma, tau, plot = FALSE, add = FALSE, main = "Estimates of DT", ...)

Arguments

data	Vector of n observations.
gamma	Vector of n-1 estimates for the EVI obtained from trMLE.
tau	Vector of n-1 estimates for the \tau obtained from trMLE.
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. GPD).

We estimate this truncation odds as

\hat{D}_T=\max\{ (k+1)/(n+1) ( (1+\hat{\tau}_k E_{1,k})^{-1/\hat{\xi}_k} - 1/(k+1) ) / (1-(1+\hat{\tau}_k E_{1,k})^{-1/\hat{\xi}_k}), 0\}

with E_{1,k} = X_{n,n}-X_{n-k,n}.

See Beirlant 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.

References

Beirlant, J., Fraga Alves, M. I. and Reynkens, T. (2017). "Fitting Tails Affected by Truncation". Electronic Journal of Statistics, 11(1), 2026–2065.

See Also

trMLE, trEndpointMLE, trProbMLE, trQuantMLE, trTestMLE, trDT

Examples

# Sample from GPD truncated at 99% quantile
gamma <- 0.5
sigma <- 1.5
X <- rtgpd(n=250, gamma=gamma, sigma=sigma, endpoint=qgpd(0.99, gamma=gamma, sigma=sigma))

# Truncated ML estimator
trmle <- trMLE(X, plot=TRUE, ylim=c(0,2))

# Truncation odds
dtmle <- trDTMLE(X, gamma=trmle$gamma, tau=trmle$tau, plot=TRUE, ylim=c(0,0.05))

TReynkens/ReIns documentation built on Nov. 9, 2023, 1:29 p.m.