Classical goodness of fit tests are no longer asymptotically distributional free if parameters are estimated. For a parametric model and the maximum likelihood estimator the empirical processes with estimated parameters is asymptotically transformed into a time transformed Brownian bridge by adding an independent Gaussian process that is suitably constructed. This randomization makes the classical tests distributional free. The power under local alternatives is investigated. Computer simulations compare the randomized Cramér-von Mises test with tests specially designed for location-scale families, such as the Shapiro-Wilk and the Shenton-Bowman test for normality and with the Epps-Pulley test for exponentiality.
\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.