trProbMLE: 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", ...)

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.

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.

Author

Tom Reynkens.

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.

References

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

Examples

Run this code

# 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))

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