trQuant function

Estimator of large quantiles using truncated Hill

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.

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).
• 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.

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.

Returns

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.

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, 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))