# Towards a better understanding of the dual representation of phi divergences

Diaa Al Mohamad: Laboratoire de Statistique Théorique et Appliquée

Statistical Papers, 2018, vol. 59, issue 3, 1205-1253

Abstract: Abstract The aim of this paper is to study different estimation procedures based on $$\varphi$$ φ -divergences. The dual representation of $$\varphi$$ φ -divergences based on the Fenchel–Legendre duality provides a way to estimate $$\varphi$$ φ -divergences by a simple plug-in of the empirical distribution without any smoothing technique. Resulting estimators are thoroughly studied theoretically and with simulations showing that the so called minimum $$\varphi$$ φ -divergence estimator is generally non robust and behaves similarly to the maximum likelihood estimator. We give some arguments supporting the non robustness property, and give insights on how to modify the classical approach. An alternative class of robust estimators based on the dual representation of $$\varphi$$ φ -divergences is introduced. We study consistency and robustness properties from an influence function point of view of the new estimator. In a second part, we invoke the Basu–Lindsay approach for approximating $$\varphi$$ φ -divergences and provide a comparison between these approaches. The so called dual $$\varphi$$ φ -divergence is also discussed and compared to our new estimator. A full simulation study of all these approaches is given in order to compare efficiency and robustness of all mentioned estimators against the so-called minimum density power divergence, showing encouraging results in favor of our new class of minimum dual $$\varphi$$ φ -divergences.

Keywords: Robust estimator; Fenchel duality; $$\varphi$$ φ -divergence; Kernel density estimation; Mixture model; 62F10; 62F35; 62F12; 62G07 (search for similar items in EconPapers)
Date: 2018
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