ReIns (version 1.0.7)

trDTMLE: 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.

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. 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.

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

# NOT RUN {
# 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))
# }