var-methods: Calculate the sample Fréchet variance of a random interval

In IntervalQuestionStat: Tools to Deal with Interval-Valued Responses in Questionnaires

Calculate the sample Fréchet variance of a random interval

Description

This function calculates the sample Fréchet variance of a single realization of n nonempty compact real intervals drawn from an interval-valued random set stored as an IntervalList object.

Usage

## S4 method for signature 'IntervalList'
var(x, theta = 1)

Arguments

x	A list of intervals, that is, an IntervalList object.
theta	A single positive real number saved as a numeric object. By default, theta = 1.

Details

Let \mathcal{X} be an interval-valued random set and let ≤ft(x_{1},x_{2},…,x_{n}\right) be a sample of n independent observations drawn from \mathcal{X}. Then, the sample Fréchet variance (see Fréchet, 1948) is defined as the following non-negative real number given by

s_{\mathcal{X}}^{2} = \frac{1}{n}∑_{i=1}^{n}d_{θ}^{2}≤ft(x_{i}, \overline{x}\right),

where θ>0 and \overline{x} denotes the sample Aumann mean of ≤ft(x_{1},x_{2},…,x_{n}\right). Due to θ-distance definition, this deviation measure can also be computed as follows,

s_{\mathcal{X}}^{2} = s_{\mathrm{mid}~\mathcal{X}}^{2}+θ\cdot s_{\mathrm{spr}~\mathcal{X}}^{2},

where

s_{\mathrm{mid}~\mathcal{X}} = \frac{1}{n}∑_{i=1}^{n} (\mathrm{mid}~x_{i} - \mathrm{mid}~\overline{x})^{2},

s_{\mathrm{spr}~\mathcal{X}} = \frac{1}{n}∑_{i=1}^{n} (\mathrm{spr}~x_{i} - \mathrm{spr}~\overline{x})^{2}.

Value

This function returns the calculated sample Fréchet variance of the given n interval, which is defined as a non-negative real number. Therefore, the output of this function is a single numeric object.

Author(s)

José García-García [e-mail]

References

Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l'institut Henri Poincaré, 10(4):215-310.

See Also

Other sample central tendency and covariance measures such as sample Aumann mean and sample covariance can be calculated through mean() and cov() functions, respectively.

Examples

## Some var() examples
list <- IntervalList(c(1, 3), c(2, 5))
var(list)
var(list, theta = 1/3)

IntervalQuestionStat documentation built on Nov. 1, 2022, 5:06 p.m.