BurrTailQuantile {AoE}R Documentation

Tail Quantile Estimation for the Burr Distribution

Description

The functions implement a small simulation study in order to assess the performance of tail quantile estimators based on random samples of the Burr distribution. The implemented estimators are the sample quantile and the Weissman estimator based on the moment estimator for the extreme-value index.

Usage

Burr.empirical(beta = 1, tau = 1, lambda = 1, n = 1000, p = 1/n,
        samples = 500, plot = TRUE)
Burr.Weissman(beta = 1, tau = 1, lambda = 1, n = 1000, p = 1/n, 
        samples = 500, k = 20, plot = TRUE)

Arguments

beta, tau, lambda Parameters of the Burr distribution. See ‘Details’.
n Sample size.
p Tail probability of the quantile to be estimated.
samples Number of samples.
k Determines the thresholds at which the Weissman estimator will be computed.
plot If TRUE, the results will be plotted.

Details

The Burr distribution is defined here by its distribution function

F(x) = 1 - (β / (β + x^tau))^λ

for x >= 0, with shape parameters λ, tau > 0 and scale parameter β > 0. The distribution is heavy-tailed with extreme-value index gamma = 1/(tau * λ). The quantile with excess probability 0 < p <= 1 is given by

Q(1-p) = (β * (p^(-1/λ) - 1))^(1/tau)

Interest is in estimating this tail quantile for small p, say of the order O(1/n), with n the sample size.

Let X_{1:n} < ... < X_{n:n} be the ascending order statistics of the sample. The aim of the functions Burr.empirical and Burr.Weissman is to compare the performance of the following two estimators of Q(1-p):

  1. The empirical tail quantile X_{i:n} with i equal to n*(1-p) rounded up.
  2. The Weissman estimate X_{n-k:n} (k / (n*p))^(hat(gamma)), where k = 1, ..., n-1 is such that X_{n-k:n} > 0, and with hat{gamma} an estimator of the extreme-value index gamma, assumed to be positive. The estimator implemented here is the Moment estimator of Dekkers et al. (1989).

If plot = TRUE, the function Burr.empirical produces a kernel density estimate of the sampling distribution of the empirical quantile estimator. For Burr.Weissman, the produced plot depends on whether k is a single number or a vector:

The use of the functions is mainly pedagogical. The following points stand out:

  1. Purely nonparametric estimation of tail quantiles is not a good idea, especially not for heavy-tailed distributions, for which tail quantiles lie “far apart”.
  2. Extreme value theory provides estimators which work reasonably well under very general assumptions, even for out-of-sample quantiles.
  3. The performance of extreme-value estimators depends on the choice of the threshold. Their sampling variance decreases but their bias increases as the number of upper order statistics used increases.
  4. The estimation uncertainty for these type of problems is quite large. Indeed, if one is not willing to rely on a parametric model, then one cannot reasonably expect a precise estimate in a region where there are no data.

Value

The functions are called mainly for their side-effect, which is to produce the plots described above. The function Burr.empirical silently returns a list with two components: Q, the true quantile, and Quantile.empirical, a vector of length samples with the estimates. The list Burr.Weissman silently returns a list with three components: the ones already mentioned and the additional component Quantile.Weissman, a matrix with at position (i,j) the Weissman quantile estimate for sample i and for k equal to k[j].

References

Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. The Annals of Statistics 17, 1833-1855.

See Also

Moment, TailQuantile, Weissman.q

Examples

# sampling distribution of empirical quantile estimator
Burr.empirical(beta = 2, tau = 2, lambda = 0.8,
        n = 1000, p = 0.001, samples = 500)

# sampling distribution of the
# Weissman quantile estimator
# based on the moment estimator
# for the extreme-value index
Burr.Weissman(beta = 2, tau = 2, lambda = 0.8,
        n = 1000, p = 0.001, samples = 500, k = 200)

# sampling distribution of the
# Weissman quantile estimator
# as a function of the threshold
Burr.Weissman(beta = 2, tau = 2, lambda = 0.8,
        n = 1000, p = 0.001, samples = 500, k = 50:400)

[Package AoE version 1.0.1 Index]