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The aim of this paper is to study different estimation procedures based on$\varphi-$divergences. The dual representation of $\varphi-$divergences basedon the Fenchel-Legendre duality is the main interest of this study. It providesa way to estimate $\varphi-$divergences by a simple plug-in of the empiricaldistribution without any smoothing technique. Resulting estimators arethoroughly studied theoretically and with simulations showing that the socalled minimum $\varphi-$divergence estimator (MD$\varphi$DE) is generally nonrobust and behaves similarly to the maximum likelihood estimator. We give somearguments supporting the non robustness property, and give insights on how tomodify the classical approach. An alternative class of $\varphi-$divergencesrobust estimators based on the dual representation is presented. We studyconsistency and robustness properties from an influence function point of viewof the new estimators. In a second part, we invoke the Basu-Lindsay approachfor approximating $\varphi-$divergences and provide a comparison between theseapproaches. The so called dual $\varphi-$divergence is also discussed andcompared to our new estimator. A full simulation study of all these approachesis given in order to compare efficiency and robustness of all mentionedestimators against the so-called minimum density power divergence, showingencouraging results in favor of our new class of minimum dual$\varphi-$divergences.

Towards a better understanding of the dual representation of phi divergences