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

Extended generalised variances, with applications

Luc Pronzato, Henry P. Wynn, and Anatoly A. Zhigljavsky

Full-text: Open access


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

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

Received: June 2015
Revised: January 2016
First available in Project Euclid: 9 May 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

design of experiments dispersion generalised variance maximum-dispersion measure optimal design quadratic entropy


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.

Export citation


  • [1] Björck, G. (1956). Distributions of positive mass, which maximize a certain generalized energy integral. Ark. Mat. 3 255–269.
  • [2] DeGroot, M.H. (1962). Uncertainty, information, and sequential experiments. Ann. Math. Statist. 33 404–419.
  • [3] Gantmacher, F. (1966). Théorie des Matrices. Paris: Dunod.
  • [4] Gini, C. (1921). Measurement of inequality of incomes. Econ. J. 31 124–126.
  • [5] Giovagnoli, A. and Wynn, H.P. (1995). Multivariate dispersion orderings. Statist. Probab. Lett. 22 325–332.
  • [6] Hainy, M., Müller, W.G. and Wynn, H.P. (2014). Learning functions and approximate Bayesian computation design: ABCD. Entropy 16 4353–4374.
  • [7] Harman, R. (2004). Lower bounds on efficiency ratios based on $\Phi_{p}$-optimal designs. In MODa 7—Advances in Model-Oriented Design and Analysis. Contrib. Statist. 89–96. Heidelberg: Physica.
  • [8] Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849–879.
  • [9] Kiefer, J. and Wolfowitz, J. (1960). The equivalence of two extremum problems. Canad. J. Math. 12 363–366.
  • [10] López-Fidalgo, J. and Rodríguez-Díaz, J.M. (1998). Characteristic polynomial criteria in optimal experimental design. In MODA 5—Advances in Model-Oriented Data Analysis and Experimental Design (Marseilles, 1998). Contrib. Statist. 31–38. Heidelberg: Physica.
  • [11] Macdonald, I.G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed. Oxford Mathematical Monographs. New York: Oxford Univ. Press.
  • [12] Marcus, M. and Minc, H. (1992). A Survey of Matrix Theory and Matrix Inequalities. New York: Dover Publications.
  • [13] Oja, H. (1983). Descriptive statistics for multivariate distributions. Statist. Probab. Lett. 1 327–332.
  • [14] Pronzato, L. (1998). On a property of the expected value of a determinant. Statist. Probab. Lett. 39 161–165.
  • [15] Pronzato, L. and Pázman, A. (2013). Design of Experiments in Nonlinear Models: Asymptotic Normality, Optimality Criteria and Small-Sample Properties. Lecture Notes in Statistics 212. New York: Springer.
  • [16] Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [17] Rao, C.R. (1982). Diversity and dissimilarity coefficients: A unified approach. Theoret. Population Biol. 21 24–43.
  • [18] Rao, C.R. (1982). Diversity: Its measurement, decomposition, apportionment and analysis. Sankhyā Ser. A 44 1–22.
  • [19] Rao, C.R. (1984). Convexity properties of entropy functions and analysis of diversity. In Inequalities in Statistics and Probability (Lincoln, Neb., 1982). Institute of Mathematical Statistics Lecture Notes—Monograph Series 5 68–77. Hayward, CA: IMS.
  • [20] Rao, C.R. (2010). Quadratic entropy and analysis of diversity. Sankhya A 72 70–80.
  • [21] Rodríguez-Díaz, J.M. and López-Fidalgo, J. (2003). A bidimensional class of optimality criteria involving $\phi_{p}$ and characteristic criteria. Statistics 37 325–334.
  • [22] Schilling, R.L., Song, R. and Vondraček, Z. (2012). Bernstein Functions: Theory and Applications, 2nd ed. de Gruyter Studies in Mathematics 37. Berlin: de Gruyter.
  • [23] Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.
  • [24] Shaked, M. (1982). Dispersive ordering of distributions. J. Appl. Probab. 19 310–320.
  • [25] Shor, N. and Berezovski, O. (1992). New algorithms for constructing optimal circumscribed and inscribed ellipsoids. Optim. Methods Softw. 1 283–299.
  • [26] Titterington, D.M. (1975). Optimal design: Some geometrical aspects of $D$-optimality. Biometrika 62 313–320.
  • [27] van der Vaart, H.R. (1965). A note on Wilks’ internal scatter. Ann. Math. Statist. 36 1308–1312.
  • [28] Wilks, S. (1932). Certain generalizations in the analysis of variance. Biometrika 24 471–494.
  • [29] Wilks, S.S. (1960). Multidimensional statistical scatter. In Contributions to Probability and Statistics 486–503. Stanford, CA: Stanford Univ. Press.
  • [30] Wilks, S.S. (1962). Mathematical Statistics. A Wiley Publication in Mathematical Statistics. New York: Wiley.