trProbMLE {ReIns}

Estimator of small exceedance probabilities using truncated MLE

Description

Computes estimates of a small exceedance probability P(X>q) using the estimates for the EVI obtained from the ML estimator adapted for upper truncation.

Usage

trProbMLE(data, gamma, tau, DT, q, plot = FALSE, add = FALSE, 
          main = "Estimates of small exceedance probability", ...)

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.

DT

Vector of n-1 estimates for the truncation odds obtained from trDTMLE.

q

The used large quantile (we estimate P(X>q) for q large).

plot

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

add

Logical indicating if the estimates should be added to an existing plot, default is FALSE.

main

Title for the plot, default is "Estimates of small exceedance probability".

...

Additional arguments for the plot function, see plot for more details.

Details

The probability is estimated as

\hat{p}_{T,k}(q) = (1+ \hat{D}_{T,k}) (k+1)/(n+1) (1+\hat\tau _k(q-X_{n-k,n}))^{-1/\hat{\xi}_k} -\hat{D}_{T,k}

with \hat{\gamma}_k and \hat{\tau}_k the ML estimates adapted for truncation and \hat{D}_T the estimates for the truncation odds.

See Beirlant et al. (2017) for more details.

Value

A list with following components:

k

Vector of the values of the tail parameter k.

P

Vector of the corresponding probability estimates.

q

The used large quantile.

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, trDTMLE, trQuantMLE, trEndpointMLE, trTestMLE, trProb, Prob

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=FALSE)

# Small exceedance probability
trProbMLE(X, gamma=trmle$gamma, tau=trmle$tau, DT=dtmle$DT, plot=TRUE, q=26, ylim=c(0,0.005))