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We study minimization of a parametric family of relative entropies, termedrelative $\alpha$-entropies (denoted $\mathscr{I}_{\alpha}(P,Q)$). These ariseas redundancies under mismatched compression when cumulants of compressedlengths are considered instead of expected compressed lengths. These parametricrelative entropies are a generalization of the usual relative entropy(Kullback-Leibler divergence). Just like relative entropy, these relative$\alpha$-entropies behave like squared Euclidean distance and satisfy thePythagorean property. Minimization of $\mathscr{I}_{\alpha}(P,Q)$ over thefirst argument on a set of probability distributions that constitutes a linearfamily is studied. Such a minimization generalizes the maximum R\'{e}nyi orTsallis entropy principle. The minimizing probability distribution (termed$\mathscr{I}_{\alpha}$-projection) for a linear family is shown to have apower-law.

Relative $\alpha$-Entropy Minimizers Subject to Linear Statistical Constraints