trQuantMLE function

Estimator of large quantiles using truncated MLE

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.

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 $\tau$ 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.

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

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

See Beirlant 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

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

Author(s)

Tom Reynkens.

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)