Hpoints: Plotting positions for exponential return levels

In Renext: Renewal Method for Extreme Values Extrapolation

Plotting positions for exponential return levels

Description

Plotting positions for exponential return level plots.

Usage

  Hpoints(n)

Arguments

n	Sample size.

Details

The plotting positions are numeric values to use as the abscissae corresponding to the order statistics in an exponential return level plot. They range from 1 to about \log n. They can be related to the plotting positions given by ppoints.

The returned vector \mathbf{H} has elements

H_{i} = \frac{1}{n} + \frac{1}{n-1} + \dots + \frac{1}{n + 1 -i}

for 1 \leq i \leq n. This is the expectation of the i-th order statistic for a sample of the standard exponential distribution, see e.g. chap. 4 of Embrechts et al.

Value

Numeric vector of plotting positions with length n.

Note

For n large enough, the largest value H_n is approximately \gamma + \log n where \gamma is the Euler-Mascheroni constant, and \exp H_n is about 1.78 n. Thus if the Hpoints are used as plotting positions on a return level plot, the largest observation has a return period of about 1.78 n years.

Author(s)

Yves Deville

References

Embrechts P., Klüppelberg C. and Mikosch T. (1997) Modelling Extremal Events for Insurance and Finance. Springer.

See Also

ppoints.

Examples

n <- 30
set.seed(1234)
x <- rGPD(n, shape = 0.2)
plot(exp(Hpoints(n)), sort(x), log = "x",
     main = "Basic return level plot")

Renext documentation built on Aug. 27, 2025, 5:13 p.m.