ReIns (version 1.0.10)

trQuant: Estimator of large quantiles using truncated Hill

Description

trQuant computes estimates of large quantiles \(Q(1-p)\) of the truncated distribution using the estimates for the EVI obtained from the Hill estimator adapted for upper truncation. trQuantW computes estimates of large quantiles \(Q_W(1-p)\) of the parent distribution \(W\) which is unobserved.

Usage

trQuant(data, r = 1, rough = TRUE, gamma, DT, p, plot = FALSE, add = FALSE, 
        main = "Estimates of extreme quantile", ...)
        
trQuantW(data, gamma, DT, p, plot = FALSE, add = FALSE, 
         main = "Estimates of extreme quantile", ...)

Value

A list with following components:

k

Vector of the values of the tail parameter \(k\).

Q

Vector of the corresponding quantile estimates.

p

The used exceedance probability.

Arguments

data

Vector of \(n\) observations (truncated data).

r

Trimming parameter, default is 1 (no trimming).

rough

Logical indicating if rough truncation is present, default is TRUE.

gamma

Vector of \(n-1\) estimates for the EVI obtained from trHill.

DT

Vector of \(n-1\) estimates for the truncation odds obtained from trDT.

p

The exceedance probability of the quantile (we estimate \(Q(1-p)\) for \(p\) small).

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 extreme quantile".

...

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

Author

Tom Reynkens based on R code of Dries Cornilly.

Details

We observe the truncated r.v. \(X=_d W | W<T\) where \(T\) is the truncation point and \(W\) the untruncated r.v.

Under rough truncation, the quantiles for \(X\) are estimated using $$\hat{Q}(1-p)=X_{n-k,n} ((\hat{D}_T + (k+1)/(n+1))/(\hat{D}_T+p))^{\hat{\gamma}_k},$$ with \(\hat{\gamma}_k\) the Hill estimates adapted for truncation and \(\hat{D}_T\) the estimates for the truncation odds.

Under light truncation, the quantiles are estimated using the Weissman estimator with the Hill estimates replaced by the truncated Hill estimates: $$\hat{Q}(1-p)=X_{n-k,n} ((k+1)/((n+1)p))^{\hat{\gamma}_k}.$$

To decide between light and rough truncation, one can use the test implemented in trTest.

The quantiles for \(W\) are estimated using $$\hat{Q}_W(1-p)=X_{n-k,n} ( (\hat{D}_T + (k+1)/(n+1)) / (p(1+\hat{D}_T))^{\hat{\gamma}_k}.$$

See Beirlant et al. (2016) or Section 4.2.3 of Albrecher et al. (2017) for more details.

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.

See Also

trHill, trDT, trProb, trEndpoint, trTest, Quant, trQuantMLE

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))

# Truncation odds
dt <- trDT(X, gamma=trh$gamma, plot=TRUE, ylim=c(0,2))

# Large quantile
p <- 10^(-5)
# Truncated distribution
trQuant(X, gamma=trh$gamma, DT=dt$DT, p=p, plot=TRUE)
# Original distribution
trQuantW(X, gamma=trh$gamma, DT=dt$DT, p=p, plot=TRUE, ylim=c(0,1000))