trProbMLE function

Estimator of small exceedance probabilities using truncated MLE

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.

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

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.

Returns

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.

References

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

Author(s)

Tom Reynkens.

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))