{
  "id": "1411.6428",
  "version": "v1",
  "published": "2014-11-24T12:32:41.000Z",
  "updated": "2014-11-24T12:32:41.000Z",
  "title": "An extended Generalised Variance, with Applications",
  "authors": [
    "Luc Pronzato",
    "Henry Wynn",
    "Anatoly Zhigljavsky"
  ],
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We consider a measure $\\psi$ k of dispersion which extends the notion of Wilk's generalised variance, or entropy, for a d-dimensional distribution, and is based on the mean squared volume of simplices of dimension k $\\le$ 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 entropy-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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-24T12:32:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extended generalised variance",
      "applications",
      "dispersion",
      "wilks generalised variance",
      "d-dimensional distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.6428P"
    }
  }
}