Weissman.q {AoE}R Documentation

Weissman Quantile Estimator

Description

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.

Usage

Weissman.q(EVI, p, plot = TRUE, ...)

Arguments

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.

Details

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).

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 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.

References

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.

See Also

Burr.Weissman, Hill, ML, Moment, TailQuantile

Examples

# 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)

[Package AoE version 1.0.1 Index]