ReIns: trQuantMLE – R documentation

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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", ...)

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.

Author(s)

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.

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)

ReIns

Functions from "Reinsurance: Actuarial and Statistical Aspects"

v1.0.10

GPL (>= 2)

Authors

Tom Reynkens [aut, cre] (<https://orcid.org/0000-0002-5516-5107>), Roel Verbelen [aut] (R code for Mixed Erlang distribution, <https://orcid.org/0000-0002-2347-9240>), Anastasios Bardoutsos [ctb] (Original R code for cEPD estimator), Dries Cornilly [ctb] (Original R code for EVT estimators for truncated data), Yuri Goegebeur [ctb] (Original S-Plus code for basic EVT estimators), Klaus Herrmann [ctb] (Original R code for GPD estimator)

Initial release

2020-05-16