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تسجيل مستخدم جديدCharacterizing Logarithmic Bregman Functions
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Minimum divergence procedures based on the density power divergence and the logarithmic density power divergence have been extremely popular and successful in generating inference procedures which combine a high degree of model efficiency with strong outlier stability. Such procedures are always preferable in practical situations over procedures which achieve their robustness at a major cost of efficiency or are highly efficient but have poor robustness properties. The density power divergence (DPD) family of Basu et al.(1998) and the logarithmic density power divergence (LDPD) family of Jones et al.(2001) provide flexible classes of divergences where the adjustment between efficiency and robustness is controlled by a single, real, non-negative parameter. The usefulness of these two families of divergences in statistical inference makes it meaningful to search for other related families of divergences in the same spirit. The DPD family is a member of the class of Bregman divergences, and the LDPD family is obtained by log transformations of the different segments of the divergences within the DPD family. Both the DPD and LDPD families lead to the Kullback-Leibler divergence in the limiting case as the tuning parameter $alpha rightarrow 0$. In this paper we study this relation in detail, and demonstrate that such log transformations can only be meaningful in the context of the DPD (or the convex generating function of the DPD) within the general fold of Bregman divergences, giving us a limit to the extent to which the search for useful divergences could be successful.
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The density power divergence (DPD) and related measures have produced many useful statistical procedures which provide a good balance between model efficiency on one hand, and outlier stability or robustness on the other. The large number of citation
s received by the original DPD paper (Basu et al., 1998) and its many demonstrated applications indicate the popularity of these divergences and the related methods of inference. The estimators that are derived from this family of divergences are all M-estimators where the defining $psi$ function is based explicitly on the form of the model density. The success of the minimum divergence estimators based on the density power divergence makes it imperative and meaningful to look for other, similar divergences in the same spirit. The logarithmic density power divergence (Jones et al., 2001), a logarithmic transform of the density power divergence, has also been very successful in producing inference procedures with a high degree of efficiency simultaneously with a high degree of robustness. This further strengthens the motivation to look for statistical divergences that are transforms of the density power divergence, or, alternatively, members of the functional density power divergence class. This note characterizes the functional density power divergence class, and thus identifies the available divergence measures within this construct that may possibly be explored for robust and efficient statistical inference.
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