Plotting positions for exponential return levels

Plotting positions for exponential return level plots.

Usage

Hpoints(n)

Arguments

n: Sample size.

Value

Numeric vector of plotting positions with length n.

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.

References

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

Author

Yves Deville

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.

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")