RiskMeasureEstimators {AoE}R Documentation

Risk Measure Estimators

Description

Estimators of various of tail-related risk measures.

Usage

EconomicCapital(y, p, k = 5:(length(y) - 1), 
        approx = "GPD", method = "ML", plot = TRUE, ...)
ExcessLoss(y, a, b = Inf, k = 5:(length(y) - 1), 
        approx = "GPD", method = "ML", plot = TRUE, ...)
Expectation(y, k = 5:(length(y) - 1), 
        approx = "GPD", method = "ML", plot = TRUE, ...)
ExpectedShortfall(y, p, k = 5:(length(y) - 1), 
        approx = "GPD", method = "ML", plot = TRUE, ...)
PHtransform(y, eta = 1, k = 5:(length(y) - 1),
        approx = "GPD", method = "ML", plot = TRUE, ...)
TailQuantile(y, p, k = 5:(length(y) - 1), 
        approx = "GPD", method = "Moment", 
        choose.k = FALSE, B = 1000, leave.out = 20, 
        k_rho = ceiling(length(y)^0.95), 
        test = "s", alpha = 0.5, plot = TRUE, ...)
Variance(y, k = 5:(length(y) - 1), 
        approx = "GPD", method = "ML", plot = TRUE, ...)

Arguments

y A numeric vector containing the data.
p Tail probability.
a,b Upper and lower limits of the excess-of-loss reinsurance contract.
eta Exponent eta in the definition of the PH-transform.
k Vector of k values, determining at which threshold(s) the estimator will be computed.
approx Approximation method for the tail: "GPD" (default) or "Weissman". The latter method is suitable only for heavy-tailed distributions, that is, with extreme-value index gamma > 0.
method Estimation method for the tail parameters: "Hill", "ML", or "Moment". Will be passed on to the function GPD_par.
plot Whether or not the results will be plotted. Defaults to TRUE.
choose.k If FALSE (the default), no automated threshold selection will be attempted. The other two possibilities are:
  • "Bootstrap", in which case k will be chosen according to the bootstrap method of Ferreira et al. (2003);
  • "Test", in which case k will be chosen according to an experimental method described in ChooseK.
B, leave.out, k_rho If choose.k = "Bootstrap", these parameters are passed on to the boostrap procedure to select k; see ‘Details’.
test, alpha If choose.k = "Test", these arguments are passed on to ChooseK for the selection of k. If plot = TRUE, then the choice of k is illustrated through a number of extra graphs, see ChooseK.
... Further arguments passed on to plot. For instance, log = "x" puts the horizontal axis on a logarithmic scale, which sometimes facilitates the choice of the threshold via k.

Details

See RiskMeasure of a description of the risk measures above.

The risk measures are estimated as functionals of the estimated distribution. The latter is estimated in two pieces:

  1. Nonparametrically up to the threshold X_{n-k:n}, the (k+1)-largest order statistic of the sample.
  2. With extreme-value theory beyond X_{n-k:n}.

The argument approx determines which approximation is used for the tail beyond X_{n-k:n}:

In both cases, the tail parameters are estimated by a call to the function GPD_par with arguments approx and method.

For the function TailQuantile, two methods for automated threshold selection are implemented:

Value

An object with class attribute "EVI", that is, a list with the following components:

n Sample size.
k Number of threshold excesses.
threshold Vector of thresholds.
estimate Vector of point estimates.
CI NULL (Confidence intervals are still to be implemented; however, for tail quantiles, see Weissman.q.)
data A character string indicating the name of the data.
quantity A character string describing the quantity being estimated.
method A character string describing the estimator.
gamma Vector of estimates of the extreme-value index.
... If choose.k is "Bootstrap" or "Test", the list contains a number of additional component providing diagnostics related to the choice of k [help file under construction].

References

A. Ferreira, L. de Haan and L. Peng (2003). On optimizing the estimation of high quantiles of a probability distribution. Statistics 37, 401-434.

Wang, S. (1995). Insurance Pricing and Increased Limits Ratemaking by Proportional Hazards Transforms. Insurance: Mathematics and Economics 17, 43-54.

Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations. Journal of the American Statistical Association 73, 812-815.

See Also

Hill, ML, Moment, GPD_par, RiskMeasure, Weissman.q

Examples

# tail quantile and excess-of-loss net premium
# for Loss data of Frees and Valdez (1998)
data(Loss)
TailQuantile(Loss, p = 0.001, k = 25:200)
ExcessLoss(Loss, a = 1.2e6, k = 25:200)

[Package AoE version 1.0.1 Index]