trQuant {ReIns}R Documentation

Estimator of large quantiles using truncated Hill

Description

trQuant computes estimates of large quantiles Q(1p)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 QW(1p)Q_W(1-p) of the parent distribution WW 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", ...)        

Arguments

data

Vector of nn 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 n1n-1 estimates for the EVI obtained from trHill.

DT

Vector of n1n-1 estimates for the truncation odds obtained from trDT.

p

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

plot

Logical indicating if the estimates should be plotted as a function of kk, 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.

Details

We observe the truncated r.v. X=dWW<TX=_d W | W<T where TT is the truncation point and WW the untruncated r.v.

Under rough truncation, the quantiles for XX are estimated using

Q^(1p)=Xnk,n((D^T+(k+1)/(n+1))/(D^T+p))γ^k,\hat{Q}(1-p)=X_{n-k,n} ((\hat{D}_T + (k+1)/(n+1))/(\hat{D}_T+p))^{\hat{\gamma}_k},

with γ^k\hat{\gamma}_k the Hill estimates adapted for truncation and D^T\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:

Q^(1p)=Xnk,n((k+1)/((n+1)p))γ^k.\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 WW are estimated using

Q^W(1p)=Xnk,n((D^T+(k+1)/(n+1))/(p(1+D^T))γ^k.\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.

Value

A list with following components:

k

Vector of the values of the tail parameter kk.

Q

Vector of the corresponding quantile estimates.

p

The used exceedance probability.

Author(s)

Tom Reynkens based on R code of Dries Cornilly.

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))


[Package ReIns version 1.0.14 Index]