## Bernoulli

• Bernoulli
• Volume 23, Number 4A (2017), 2617-2642.

### Extended generalised variances, with applications

#### Abstract

We consider a measure $\psi_{k}$ of dispersion which extends the notion of Wilk’s generalised variance for a $d$-dimensional distribution, and is based on the mean squared volume of simplices of dimension $k\leq d$ formed by $k+1$ independent copies. We show how $\psi_{k}$ can be expressed in terms of the eigenvalues of the covariance matrix of the distribution, also when a $n$-point sample is used for its estimation, and prove its concavity when raised at a suitable power. Some properties of dispersion-maximising distributions are derived, including a necessary and sufficient condition for optimality. Finally, we show how this measure of dispersion can be used for the design of optimal experiments, with equivalence to $A$ and $D$-optimal design for $k=1$ and $k=d$, respectively. Simple illustrative examples are presented.

#### Article information

Source
Bernoulli, Volume 23, Number 4A (2017), 2617-2642.

Dates
Revised: January 2016
First available in Project Euclid: 9 May 2017

https://projecteuclid.org/euclid.bj/1494316827

Digital Object Identifier
doi:10.3150/16-BEJ821

Mathematical Reviews number (MathSciNet)
MR3648040

Zentralblatt MATH identifier
06778251

#### Citation

Pronzato, Luc; Wynn, Henry P.; Zhigljavsky, Anatoly A. Extended generalised variances, with applications. Bernoulli 23 (2017), no. 4A, 2617--2642. doi:10.3150/16-BEJ821. https://projecteuclid.org/euclid.bj/1494316827

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