# trQuantMLE: Estimator of large quantiles using truncated MLE In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

## Description

This function computes estimates of large quantiles Q(1-p) of the truncated distribution using the ML estimates adapted for upper truncation. Moreover, estimates of large quantiles Q_Y(1-p) of the original distribution Y, which is unobserved, are also computed.

## Usage

 1 2 trQuantMLE(data, gamma, tau, DT, p, Y = FALSE, plot = FALSE, add = FALSE, main = "Estimates of extreme quantile", ...) 

## Arguments

 data Vector of n observations. gamma Vector of n-1 estimates for the EVI obtained from trMLE. tau Vector of n-1 estimates for the τ obtained from trMLE. DT Vector of n-1 estimates for the truncation odds obtained from trDTMLE. p The exceedance probability of the quantile (we estimate Q(1-p) or Q_Y(1-p) for p small). Y Logical indicating if quantiles from the truncated distribution (Q(1-p)) or from the parent distribution (Q_Y(1-p)) are computed. 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 extreme quantile". ... Additional arguments for the plot function, see plot for more details.

## Details

We observe the truncated r.v. X=_d Y | Y<T where T is the truncation point and Y the untruncated r.v.

Under rough truncation, the quantiles for X are estimated using

\hat{Q}_{T,k}(1-p) = X_{n-k,n} +1/(\hat{τ}_k)([(\hat{D}_{T,k} + (k+1)/(n+1))/(\hat{D}_{T,k}+p)]^{\hat{ξ}_k} -1),

with \hat{γ}_k and \hat{τ}_k the ML estimates adapted for truncation and \hat{D}_T the estimates for the truncation odds.

The quantiles for Y are estimated using

\hat{Q}_{Y,k}(1-p)=X_{n-k,n} +1/(\hat{τ}_k)([(\hat{D}_{T,k} + (k+1)/(n+1))/(p(\hat{D}_{T,k}+1))]^{\hat{ξ}_k} -1).

See Beirlant et al. (2017) for more details.

## 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.

Tom Reynkens.

## References

Beirlant, J., Fraga Alves, M. I. and Reynkens, T. (2017). "Fitting Tails Affected by Truncation". Electronic Journal of Statistics, 11(1), 2026–2065.

trMLE, trDTMLE, trProbMLE, trEndpointMLE, trTestMLE, trQuant, Quant
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 # Sample from GPD truncated at 99% quantile gamma <- 0.5 sigma <- 1.5 X <- rtgpd(n=250, gamma=gamma, sigma=sigma, endpoint=qgpd(0.99, gamma=gamma, sigma=sigma)) # Truncated ML estimator trmle <- trMLE(X, plot=TRUE, ylim=c(0,2)) # Truncation odds dtmle <- trDTMLE(X, gamma=trmle$gamma, tau=trmle$tau, plot=FALSE) # Large quantile of X trQuantMLE(X, gamma=trmle$gamma, tau=trmle$tau, DT=dtmle$DT, plot=TRUE, p=0.005, ylim=c(15,30)) # Large quantile of Y trQuantMLE(X, gamma=trmle$gamma, tau=trmle$tau, DT=dtmle$DT, plot=TRUE, p=0.005, ylim=c(0,300), Y=TRUE)