ReIns (version 1.0.7)

MeanExcess_TB: Mean excess function using Turnbull estimator

Description

Computes mean excess values using the Turnbull estimator. These mean excess values can then be plotted as a function of the empirical quantiles (computed using the Turnbull estimator) or as a function of the tail parameter \(k\).

Usage

MeanExcess_TB(L, U = L, censored, trunclower = 0, truncupper = Inf, 
              plot = TRUE, k = FALSE, intervalpkg = TRUE, 
              main = "Mean excess plot", ...)

Arguments

L

Vector of length \(n\) with the lower boundaries of the intervals for interval censored data or the observed data for right censored data.

U

Vector of length \(n\) with the upper boundaries of the intervals. By default, they are equal to L.

censored

A logical vector of length \(n\) indicating if an observation is censored.

trunclower

Lower truncation point, default is 0.

truncupper

Upper truncation point, default is Inf.

plot

Logical indicating if the mean excess values should be plotted in a mean excess plot, default is TRUE.

k

Logical indicating if the mean excess values are plotted as a function of the tail parameter \(k\) (k=TRUE) or as a function of the empirical quantiles computed using the Turnbull estimator (k=FALSE). Default is FALSE.

intervalpkg

Logical indicating if the Turnbull estimator is computed using the implementation in the interval package if this package is installed. Default is TRUE.

main

Title for the plot, default is "Mean excess plot".

…

Additional arguments for the plot function, see plot for more details.

Value

A list with following components:

k

Vector of the values of the tail parameter k.

X

Vector of the empirical quantiles, computed using the Turnbull estimator, corresponding to (n-k)/(n+1)=1-(k+1)/(n+1).

e

Vector of the mean excess values corresponding to the tail parameters in k.

Details

The mean excess values are given by $$\hat{e}^{TB}(v)=(\int_v^{\infty} 1-\hat{F}^{TB}(u) du)/(1-\hat{F}^{TB}(v))$$ where \(\hat{F}^{TB}\) is the Turnbull estimator for the CDF. More specifically, we use the values \(v=\hat{Q}^{TB}(p)\) for \(p=1/(n+1), \ldots, (n-1)/(n+1)\) where \(\hat{Q}^{TB}(p)\) is the empirical quantile function corresponding to the Turnbull estimator.

Right censored data should be entered as L=l and U=truncupper, and left censored data should be entered as L=trunclower and U=u.

If the interval package is installed and intervalpkg=TRUE, the icfit function is used to compute the Turnbull estimator. Otherwise, survfit.formula from survival is used.

Use MeanExcess for non-censored data.

See Section 4.3 in Albrecher et al. (2017) for more details.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

See Also

MeanExcess, Turnbull, icfit

Examples

# NOT RUN {
# Pareto random sample
X <- rpareto(500, shape=2)

# Censoring variable
Y <- rpareto(500, shape=1)

# Observed sample
Z <- pmin(X, Y)

# Censoring indicator
censored <- (X>Y)

# Right boundary
U <- Z
U[censored] <- Inf

# Mean excess plot
MeanExcess_TB(Z, U, censored, k=FALSE)
# }