MeanExcess_TB function

Mean excess function using Turnbull estimator

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 $k$.

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.

Details

The mean excess values are given by

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.

Returns

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.

References

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

Author(s)

Tom Reynkens

See Also

MeanExcess, Turnbull, icfit

Examples

# 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)