ReIns: MeanExcess_TB – R documentation

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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.

Details

The mean excess values are given by

\hat{e}^{TB}(v)=(\int_v^{∞} 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), …, (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.

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`.

Author(s)

Tom Reynkens

References

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

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)

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