  Tsallis and R\'{e}nyi entropies, which are monotone transformations of each
other, are deformations of the celebrated Shannon entropy. Maximization of
these deformed entropies, under suitable constraints, leads to the
$q$-exponential family which has applications in non-extensive statistical
physics, information theory and statistics. In previous information-geometric
studies, the $q$-exponential family was analyzed using classical convex duality
and Bregman divergence. In this paper, we show that a generalized
$\lambda$-duality, where $\lambda = 1 - q$ is the constant
information-geometric curvature, leads to a generalized exponential family
which is essentially equivalent to the $q$-exponential family and has deep
connections with R\'{e}nyi entropy and optimal transport. Using this
generalized convex duality and its associated logarithmic divergence, we show
that our $\lambda$-exponential family satisfies properties that parallel and
generalize those of the exponential family. Under our framework, the R\'{e}nyi
entropy and divergence arise naturally, and we give a new proof of the
Tsallis/R\'{e}nyi entropy maximizing property of the $q$-exponential family. We
also introduce a $\lambda$-mixture family which may be regarded as the dual of
the $\lambda$-exponential family, and connect it with other mixture-type
families. Finally, we discuss a duality between the $\lambda$-exponential
family and the $\lambda$-logarithmic divergence, and study its statistical
consequences.
