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", ...)

Arguments

data

Vector of $n$ observations (truncated data).

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.

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

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.

Value

A list with following components:

Vector of the values of the tail parameter $k$ .

Vector of the corresponding quantile estimates.

The used exceedance probability.

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 $\hat{Q} (1 - p) = X_{n - k, n} (({\hat{D}}_{T} + (k + 1) / (n + 1)) / ({\hat{D}}_{T} + p))^{{\hat{γ}}_{k}},$ with ${\hat{γ}}_{k}$ 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: $\hat{Q} (1 - p) = X_{n - k, n} ((k + 1) / ((n + 1) p))^{{\hat{γ}}_{k}} .$

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

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

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.

Examples

Run this code

# NOT RUN {
# 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))

# }

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