RiskMeasureEstimators {AoE} | R Documentation |
Estimators of various of tail-related risk measures.
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, ...)
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:
|
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. |
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:
The argument approx
determines which approximation is used for the tail beyond X_{n-k:n}:
approx = "Weissman"
, the tail probability 1 - F(x) at x > X_{n-k:n} > 0 is estimated as
(k/n) * (x/X_{n-k:n})^gamma
with gamma a positive estimator of the extreme-value index.
approx = "GPD"
, the tail probability 1 - F(x) at x > X_{n-k:n} is estimated as
(k/n) * (1 + gamma * (x - X_{n-k:n}) / σ)^(-1/gamma)
with gamma and σ estimators of the generalized Pareto distribution fitted to the excesses over the threshold 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:
choose.k = "Bootstrap"
, the bootstrap method of Ferreira et al. (2003). This method is implemented in the function TQ_ChooseK
[help file under construction], to which the additional arguments are passed on:
B
, the number of bootstrap samples;
leave.out
, the number of lowest and highest k-values to leave out in the search for the k that minimizes the estimated asymptotic mean squared error;
k_rho
, determining the threshold at which to estimate the second-order parameter \rho
.
choose.k = "Test"
, an experimental method implemented in the function ChooseK
and with arguments test
and alpha
.
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]. |
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.
Hill
, ML
, Moment
, GPD_par
, RiskMeasure
, Weissman.q
# 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)