BurrTailQuantile {AoE} | R Documentation |
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.
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)
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. |
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):
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:
k
is a single number, the plot shows a kernel density estimate of the sampling distribution of the Weissman quantile estimator.
k
is a vector, the plot shows the estimated 5/50/95 percentiles of the sampling distribution of the Weissman quantile estimator as a function of k. For comparison, the corresponding percentiles for the empirical quantile estimator are shown as well.
The use of the functions is mainly pedagogical. The following points stand out:
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]
.
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.
Moment
, TailQuantile
, Weissman.q
# 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)