trProb function

Estimator of small exceedance probabilities using truncated Hill

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

trProb(data, r = 1, gamma, q, warnings = TRUE, plot = FALSE, add = FALSE, 
       main = "Estimates of small exceedance probability", ...)

Arguments

• data: Vector of $n$ observations.
• r: Trimming parameter, default is 1 (no trimming).
• gamma: Vector of $n-1$ estimates for the EVI obtained from trHill.
• q: The used large quantile (we estimate $P(X>q)$ for $q$ large).
• warnings: Logical indicating if warnings are shown, default is TRUE.
• 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 $R_{r,k,n} = X_{n-k,n} / X_{n-r+1,n}$ and $\hat{\gamma}_k$ the Hill estimates adapted for truncation.

See Beirlant et al. (2016) or Section 4.2.3 of Albrecher 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

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant, J., Fraga Alves, M.I. and Gomes, M.I. (2016). "Tail fitting for Truncated and Non-truncated Pareto-type Distributions." Extremes, 19, 429--462.

Author(s)

Tom Reynkens based on R code of Dries Cornilly.

See Also

trHill, trQuant, Prob, trProbMLE

Examples

# Sample from truncated Pareto distribution.
# truncated at 99% quantile
shape <- 2
X <- rtpareto(n=1000, shape=shape, endpoint=qpareto(0.99, shape=shape))

# Truncated Hill estimator
trh <- trHill(X, plot=TRUE, ylim=c(0,2))

# Small probability
trProb(X, gamma=trh$gamma, q=8, plot=TRUE)