Weissman.q {AoE} | R Documentation |
This function is an implementation of the Weissman (1978) estimator for a high tail quantile of a heavy-tailed distribution based on an estimate of the (positive) extreme-value index.
Weissman.q(EVI, p, plot = TRUE, ...)
EVI |
An object with class attribute "EVI" , i.e. the output of one of the functions Hill , ML , or Moment . See ‘Examples’ below. |
p |
Tail probability of the quantile to be estimated. |
plot |
If TRUE (the default), the result will be plotted. |
... |
Further arguments passed on to plot provided plot = TRUE . For instance, log = "x" , log = "y" , and log = "xy" draw the horizontal and/or the vertical axis on logarithmic scale. The former is useful for selecting k, the latter is useful for estimating extreme quantiles of very heavy-tailed distributions. |
Let X_{1:n} <= ... <= X_{n:n} be the ascending order statistics of a sample and let k = 1, ..., n-1 be such that X_{n-k:n} > 0. For 0 < p <= k/n, the Weissman (1978) estimator of the tail quantile Q(1-p) is defined as
X_{n-k:n} * (k / (n*p))^gamma
where gamma is a positive estimate of the extreme-value index.
For random samples from a distribution with positive extreme-value index and if
the Weissman quantile estimator is consistent in the sense that the relative error tends to zero. Under additional assumptions, the estimator is also asymptotically normal; see for instance Beirlant et al. (2004, section 4.6.1).
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 |
Matrix with upper and lower endpoints of confidence intervals. |
CI.type |
A character string indicating the type of confidence interval. |
CI.p |
Nominal coverage probability of confidence interval. |
data |
A character string indicating the name of the data. |
quantity |
A list with two components: name , equal to "Q" , and par , equal to 1-p . |
method |
A character string describing the estimator. |
Beirlant, J., Goegebeur, Y., Segers, J., and Teugels, J. (2004). Statistics of Extremes. Wiley, Chichester.
Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations. Journal of the American Statistical Association 73, 812-815.
Burr.Weissman
, Hill
, ML
, Moment
, TailQuantile
# norwegian fire insurance data: data(norwegian) # estimate gamma by the moment estimator: gamma.M <- Moment(norwegian$y81, CI.p = 0.9, k = 10:100) # use output gamma.M as input for the Weissman estimator: Q.M <- Weissman.q(gamma.M, p = 0.01)