extremevalueindex {AoE}R Documentation

Estimators for the extreme-value index

Description

Implementation of the Hill (1975) estimator, the moment estimator of Dekkers, Einmahl and de Haan (1989), and the maximum likelihood estimator of Smith (1987) of the extreme-value index. Allows for computation of confidence intervals and an experimental way to choose the threshold.

Usage

Hill(data, k = 5:(sum(data > 0) - 1), plot = TRUE,
        CI.type = c("Wald", "score", "none"), CI.p = NULL,
        choose.k = FALSE, test = "s", alpha = 0.5, ...)
Moment(data, k = 5:(sum(data > 0) - 1), plot = TRUE,
        CI.type = "Wald", CI.p = NULL,
        choose.k = FALSE, test = "s", alpha = 0.5, ...)
ML( data, k = 5:(length(data) - 1), plot = TRUE,
        CI.type = "Wald", CI.p = NULL,
        choose.k = FALSE, test = "s", alpha = 0.5, ...)

Arguments

data The data vector
k Vector of k values.
plot Whether or not the results will be plotted. Defaults to TRUE.
CI.type Type of confidence interval. For the functions Moment and ML, only the "Wald" symmetric confidence intervals are implemented.
CI.p Nominal coverage probability of the confidence interval. If NULL, no confidence interval will be computed.
choose.k Whether or not a choice for k will be suggested. Defaults to FALSE. See ChooseK.
test If choose.k = TRUE, determines the test with which the goodness-of-fit of the exponential distribution to the residuals will be tested. See ChooseK.
alpha If choose.k = TRUE, determines the nominal size of the test with which the goodness-of-fit of the exponential distribution to the log-excesses will be tested. See ChooseK.
... Further arguments passed on to plot.

Details

By definition, the Hill estimator always returns a positive estimate, whereas the Moment and ML estimates can have either sign.

The Wald confidence interval is the usual symmetric interval centered around the estimator and based upon the estimated standard error and the normal approximation. In case of the Hill estimator, one can also choose the more accurate, asymmetric score confidence intervals (Haeusler and Segers, 2007).

For the Hill estimator and Moment estimator, values for k for which the k+1th order statistic is nonpositive will be ignored.

If plot = TRUE, the estimates are displayed as a function of k. Add the extra argument log = "x" to display the horizontal axis on a log-scale, as in the altHill plot (Resnick and Starica 1997; Drees et al. 2000).

If choose.k = TRUE, a value of k is selected using the function ChooseK with arguments test and alpha, and with argument approx = Weissman for the function Hill and with the arguments approx = "GPD", method = "ML" and approx = "GPD", method = "Moment" for the functions ML and Moment, respectively. In addition, if plot = TRUE, three plots are shown:

Value

A list with the class attribute "EVI". If choose.k = TRUE, then the list gets the extra class attribute "ChooseK" and a number of extra fields related to the choice of k as in ChooseK.

n The sample size.
k Vector of values of k for which the Hill estimator has been computed.
threshold Vector of thresholds corresponding to k.
estimate Vector of corresponding Hill estimates.
CI Matrix of upper and lower bounds of corresponding corresponding intervals.
CI.type The type of confidence interval.
CI.p Nominal coverage probability of the confidence intervals.
std.err Vector of estimated standard errors.
data The data
quantity The string "gamma".
method The name of the estimator used: "Hill", "ML" or "Moment".
... If choose.k = TRUE, then the fields as in the output of ChooseK are present as well.

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.

Drees, H., de Haan, L., and Resnick, S. (2000). How to make a Hill plot. The Annals of Statistics 28, 254-274.

Haeusler, E. and Segers, J. (2007). Assessing confidence intervals for the tail index by Edgeworth expansions for the Hill estimator. Bernoulli 13, 175-194.

Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3, 1163-1174.

Resnick, S. and Starica, C. (1997). Smoothing the Hill estimator. Advances in Applied Probability 29, 271-293.

Smith, R.L. (1987). Estimating tails of probability distributions. The Annals of Statistics 15, 1174-1207.

See Also

ChooseK, GPD_par, Hill.diagnostic

Examples

# 1. power law with gamma = 1
x <- 1/runif(100)
Hill(x, CI.p = 0.95)
abline(h = 1, col = "blue")

# 2. altHill plot:
# display k on log-scale
# emphasise smaller k
x <- rburr(n = 1000, gamma = 0.5, rho = -0.75)
# linear scale
Hill(x, k = 10:500)
abline(h = 0.5, col = "blue")
# log scale
Hill(x, k = 10:500, log = "x")
abline(h = 0.5, col = "blue")

# 3. ML and Moment estimators
data(soa)
Moment(soa, k = 20:3000)
ML(soa, k = 20:3000)

# 4. choosing k
data(soa)
Moment(soa, k = 20:10000)
Moment(soa, k = 20:10000, choose.k = TRUE)

[Package AoE version 1.0.1 Index]