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Multivariate log-concave distributions as a nearly parametric model

Author

Listed:
  • Schuhmacher Dominic
  • Hüsler André

    (University of Bern, Institute of Mathematical Statistics and Actuarial, Bern, Schweiz)

  • Dümbgen Lutz

    (University of Bern, Institute of Mathematical Statistics and Actuarial, Bern, Schweiz)

Abstract

In this paper we show that the family Pd(lc) of probability distributions on ℝd with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. In this and several other respects the nonparametric model Pd(lc) behaves like a parametric model such as, for instance, the family of all d-variate Gaussian distributions. As a consequence of the continuity result, we prove the existence of nontrivial confidence sets for the moments of an unknown distribution in Pd(lc). Our results are based on various new inequalities for log-concave distributions which are of independent interest.

Suggested Citation

  • Schuhmacher Dominic & Hüsler André & Dümbgen Lutz, 2011. "Multivariate log-concave distributions as a nearly parametric model," Statistics & Risk Modeling, De Gruyter, vol. 28(3), pages 277-295, September.
    • Handle: RePEc:bpj:strimo:v:28:y:2011:i:3:p:277-295:n:1
    DOI: 10.1524/stnd.2011.1073
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    References listed on IDEAS

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    1. Mark Bagnoli & Ted Bergstrom, 2006. "Log-concave probability and its applications," Studies in Economic Theory, in: Charalambos D. Aliprantis & Rosa L. Matzkin & Daniel L. McFadden & James C. Moore & Nicholas C. Yann (ed.), Rationality and Equilibrium, pages 217-241, Springer.
    2. Cule, Madeleine & Gramacy, Robert B. & Samworth, Richard, 2009. "LogConcDEAD: An R Package for Maximum Likelihood Estimation of a Multivariate Log-Concave Density," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 29(i02).
    3. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
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