IntervalQuestionStat (version 0.2.0)

mean: Calculate the sample Aumann mean of a random interval

Description

This function calculates the sample Aumann mean of a single realization formed by \(n\) nonempty compact real intervals drawn from a random interval saved as an IntervalList object.

Usage

# S4 method for IntervalList
mean(x)

Value

This function returns an IntervalData object with the calculated sample Aumann mean of the given \(n\) intervals, which is defined as another nonempty compact real interval.

Arguments

x

A list of intervals, that is, an IntervalList object.

Author

José García-García

Details

Let \(\mathcal{X}\) be an interval-valued random set and let \(\left(x_{1},x_{2},\ldots,x_{n}\right)\) be a sample of \(n\) independent observations drawn from \(\mathcal{X}\). Then, the sample Aumann mean (see Aumann, 1965) is defined as the following interval given by $$\overline{x} = \frac{1}{n}\sum_{i=1}^{n} x_{i}.$$

References

Aumann, R.J. (1965). Integrals of set-valued functions. Journal of Mathematical Analysis and Applications, 12(1):1-12. tools:::Rd_expr_doi("10.1016/0022-247X(65)90049-1").

See Also

Other sample dispersion and covariance measures such as sample Fréchet variance and sample covariance can be calculated through var() and cov() functions, respectively.

Examples

## Some mean() trivial examples
list <- IntervalList(c(1, 3), c(2, 5))
mean(list)