We propose a simple method of construction of new families of ϕ-divergences. This method called convex standardization is applicable to convex and concave functions ψ(t) twice continuously differentiable in a neighborhood of t=1 with nonzero second derivative at the point t=1. Using this method we introduce several extensions of the LeCam, power, χa and Matusita divergences. The extended families are shown to connect smoothly these divergences with the Kullback divergence or they connect various pairs of these particular divergences themselves. We investigate also the metric properties of divergences from these extended families.
\noindent We introduce new estimates and tests of independence in copula models with unknown margins using ϕ-divergences and the duality technique. The asymptotic laws of the estimates and the test statistics are established both when the parameter is an interior or a boundary value of the parameter space. Simulation results show that the choice of χ2-divergence has good properties in terms of efficiency-robustness.
\vspace{-1.6cm} The paper studies the relations between ϕ-divergences and fundamental concepts of decision theory such as sufficiency, Bayes sufficiency, and LeCam's deficiency. A new and considerably simplified approach is given to the spectral representation of ϕ-divergences already established in Österreicher and Feldman \cite{OestFeld} under restrictive conditions and in Liese and Vajda \cite{LiV06}, \cite{LiV08} in the general form. The simplification is achieved by a new integral representation of convex functions in terms of elementary convex functions which are strictly convex at one point only. Bayes sufficiency is characterized with the help of a binary model that consists of the joint distribution and the product of the marginal distributions of the observation and the parameter, respectively. LeCam's deficiency is expressed in terms of ϕ-divergences where ϕ belongs to a class of convex functions whose curvature measures are finite and satisfy a normalization condition.