NMOF (version 2.10-0)

pm: Partial Moments

Description

Compute partial moments.

Usage

pm(x, xp = 2, threshold = 0, lower = TRUE,
     normalise = FALSE, na.rm = FALSE)

Value

numeric

Arguments

x

a numeric vector or a matrix

xp

exponent

threshold

a numeric vector of length one

lower

logical

normalise

logical

na.rm

logical

Author

Enrico Schumann

Details

For a vector \(x\) of length \(n\), partial moments are computed as follows:

$$\mathrm{upper\ partial\ moment} = \frac{1}{n} \sum_{x > t}\left(x - t \right)^e$$

$$\mathrm{lower\ partial\ moment} = \frac{1}{n} \sum_{x < t}\left(t - x \right)^e$$

The threshold is denoted \(t\), the exponent xp is labelled \(e\).

If normalise is TRUE, the result is raised to 1/xp. If x is a matrix, the function will compute the partial moments column-wise.

See Gilli, Maringer and Schumann (2019), chapter 14.

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). https://enricoschumann.net/NMOF.htm#NMOFmanual

Examples

pm(x <- rnorm(100), 2)
var(x)/2

pm(x, 2, normalise = TRUE)
sqrt(var(x)/2)