Hill {qrmtools}

Hill Estimator and Plot

Description

Compute the Hill estimator and Hill plot.

Usage

Hill_estimator(x, k = c(10, length(x)), conf.level = 0.95)
Hill_plot(x, k = c(10, length(x)), conf.level = 0.95, Hill.estimator = NULL,
          log = "x", xlim = NULL, ylim = NULL,
          xlab = "Order statistics", ylab = "Tail index",
          CI.col = adjustcolor(1, alpha.f = 0.2), lines.args = list(),
          xaxis2 = TRUE, xlab2 = "Empirical probability", ...)

Arguments

x

vector of numeric data.

k

vector of length 2, determining the smallest and largest number of order statistics of x to compute the Hill estimator for (the smallest needs to be >= 2). If k is of length 1, k is expanded by length(x).

conf.level

confidence level of the confidence intervals.

Hill.estimator

object as returned by Hill_estimator().

log, xlim, ylim, xlab, ylab

see plot().

CI.col

color of the pointwise asymptotic confidence intervals (CIs); if NA, no CIs are shown.

lines.args

list of additional arguments for the underlying lines() call to draw the Hill estimator.

xaxis2

logical indicating whether a third axis is drawn that shows the empirical probabilities 1-(k-1)/length(x) corresponding to k, so the value of the column k.prob as returned by Hill_estimator().

xlab2

label of the secondary x-axis.

...

additional arguments passed to the underlying plot().

Details

See McNeil et al. (2015, Section 5.2.4, (5.23))

Value

Hill_estimator():

A five-column matrix containing the indices k, their corresponding empirical probabilities k.prob, the estimated tail indices tail.index, and the lower and upper CI endpoints CI.low and CI.up.

Hill_plot():

Hill plot by side-effect.

Author(s)

Marius Hofert

References

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

Examples

set.seed(271)
X <- rt(1000, df = 3.5)
Y <- X[X > 0]
Hill_plot(Y)
Hill_plot(Y, log = "", CI.col = NA)