ReIns (version 1.0.10)

trQuantMLE: Estimator of large quantiles using truncated MLE

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

trQuantMLE(data, gamma, tau, DT, p, Y = FALSE, 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.

gamma

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

tau

Vector of \(n-1\) estimates for the \(\tau\) 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.

Author

Tom Reynkens.

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 $$\hat{Q}_{T,k}(1-p) = X_{n-k,n} +1/(\hat{\tau}_k)([(\hat{D}_{T,k} + (k+1)/(n+1))/(\hat{D}_{T,k}+p)]^{\hat{\xi}_k} -1),$$ with \(\hat{\gamma}_k\) and \(\hat{\tau}_k\) the ML estimates adapted for truncation and \(\hat{D}_T\) the estimates for the truncation odds.

The quantiles for \(Y\) are estimated using $$\hat{Q}_{Y,k}(1-p)=X_{n-k,n} +1/(\hat{\tau}_k)([(\hat{D}_{T,k} + (k+1)/(n+1))/(p(\hat{D}_{T,k}+1))]^{\hat{\xi}_k} -1).$$

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

References

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

See Also

trMLE, trDTMLE, trProbMLE, trEndpointMLE, trTestMLE, trQuant, Quant

Examples

# 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)