Modified power divergence estimators in normal models – simulation and comparative study

Iva Frýdlová; Igor Vajda; Václav Kůs

Kybernetika (2012)

  • Volume: 48, Issue: 4, page 795-808
  • ISSN: 0023-5954

Abstract

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Point estimators based on minimization of information-theoretic divergences between empirical and hypothetical distribution induce a problem when working with continuous families which are measure-theoretically orthogonal with the family of empirical distributions. In this case, the φ -divergence is always equal to its upper bound, and the minimum φ -divergence estimates are trivial. Broniatowski and Vajda [3] proposed several modifications of the minimum divergence rule to provide a solution to the above mentioned problem. We examine these new estimation methods with respect to consistency, robustness and efficiency through an extended simulation study. We focus on the well-known family of power divergences parametrized by α ∈ ℝ in the Gaussian model, and we perform a comparative computer simulation for several randomly selected contaminated and uncontaminated data sets, different sample sizes and different φ -divergence parameters.

How to cite

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Frýdlová, Iva, Vajda, Igor, and Kůs, Václav. "Modified power divergence estimators in normal models – simulation and comparative study." Kybernetika 48.4 (2012): 795-808. <http://eudml.org/doc/246594>.

@article{Frýdlová2012,
abstract = {Point estimators based on minimization of information-theoretic divergences between empirical and hypothetical distribution induce a problem when working with continuous families which are measure-theoretically orthogonal with the family of empirical distributions. In this case, the $\phi $-divergence is always equal to its upper bound, and the minimum $\phi $-divergence estimates are trivial. Broniatowski and Vajda [3] proposed several modifications of the minimum divergence rule to provide a solution to the above mentioned problem. We examine these new estimation methods with respect to consistency, robustness and efficiency through an extended simulation study. We focus on the well-known family of power divergences parametrized by $\alpha \in \mathbb \{R\}$ in the Gaussian model, and we perform a comparative computer simulation for several randomly selected contaminated and uncontaminated data sets, different sample sizes and different $\phi $-divergence parameters.},
author = {Frýdlová, Iva, Vajda, Igor, Kůs, Václav},
journal = {Kybernetika},
keywords = {minimum $\phi $-divergence estimation; subdivergence; superdivergence; PC simulation; relative efficiency; robustness; robustness; minimum -divergence estimation; subdivergence; superdivergence; PC simulation; relative efficiency},
language = {eng},
number = {4},
pages = {795-808},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Modified power divergence estimators in normal models – simulation and comparative study},
url = {http://eudml.org/doc/246594},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Frýdlová, Iva
AU - Vajda, Igor
AU - Kůs, Václav
TI - Modified power divergence estimators in normal models – simulation and comparative study
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 4
SP - 795
EP - 808
AB - Point estimators based on minimization of information-theoretic divergences between empirical and hypothetical distribution induce a problem when working with continuous families which are measure-theoretically orthogonal with the family of empirical distributions. In this case, the $\phi $-divergence is always equal to its upper bound, and the minimum $\phi $-divergence estimates are trivial. Broniatowski and Vajda [3] proposed several modifications of the minimum divergence rule to provide a solution to the above mentioned problem. We examine these new estimation methods with respect to consistency, robustness and efficiency through an extended simulation study. We focus on the well-known family of power divergences parametrized by $\alpha \in \mathbb {R}$ in the Gaussian model, and we perform a comparative computer simulation for several randomly selected contaminated and uncontaminated data sets, different sample sizes and different $\phi $-divergence parameters.
LA - eng
KW - minimum $\phi $-divergence estimation; subdivergence; superdivergence; PC simulation; relative efficiency; robustness; robustness; minimum -divergence estimation; subdivergence; superdivergence; PC simulation; relative efficiency
UR - http://eudml.org/doc/246594
ER -

References

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  1. M. Broniatowski, A. Keziou, Minimization of φ -divergences on sets of signed measures., Studia Sci. Math. Hungar. 43 (2006), 403-442. Zbl1121.28004MR2273419
  2. M. Broniatowski, A. Keziou, 10.1016/j.jmva.2008.03.011, J. Multivariate Anal. 100 (2009), 16-36. Zbl1151.62023MR2460474DOI10.1016/j.jmva.2008.03.011
  3. M. Broniatowski, I. Vajda, Several Applications of Divergence Criteria in Continuous Families., Research Report No. 2257. Institute of Information Theory and Automation, Prague 2009. 
  4. I. Frýdlová, Minimum Kolmogorov Distance Estimators., Diploma Thesis. Czech Technical University, Prague 2004. 
  5. I. Frýdlová, Modified Power Divergence Estimators and Their Performances in Normal Models., In: Proc. FernStat2010, Faculty of Social and Economic Studies UJEP, Ústí n. L. 2010, 28-33. 
  6. F. Liese, I. Vajda, 10.1109/TIT.2006.881731, IEEE Trans. Inform. Theory 52 (2006), 4394-4412. MR2300826DOI10.1109/TIT.2006.881731
  7. A. Toma, S. Leoni-Aubin, 10.1016/j.jmva.2009.11.001, J. Multivariate Anal. 101 (2010), 1143-1155. Zbl1185.62042MR2595297DOI10.1016/j.jmva.2009.11.001
  8. A. Toma, M. Broniatowski, 10.1016/j.jmva.2010.07.010, J. Multivariate Analysis 102 (2011), 20-36. Zbl1206.62034MR2729417DOI10.1016/j.jmva.2010.07.010
  9. I. Vajda, Theory of Statistical Inference and Information., Kluwer, Boston 1989. Zbl0711.62002

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