Hill function

Hill Estimator and Plot

Compute the Hill estimator and Hill plot.

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().

Returns

• 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.

Details

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

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)