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By extrapolating the explicit formula of the zero-bias distribution occurringin the context of Stein's method, we construct characterization identities fora large class of absolutely continuous univariate distributions. Instead oftrying to derive characterizing distributional transformations that inheritcertain structures for the use in further theoretic endeavours, we focus onexplicit representations given through a formula for the density- ordistribution function. The results we establish with this ambition featureimmediate applications in the area of goodness-of-fit testing. We draw up ablueprint for the construction of tests of fit that include procedures for manydistributions for which little (if any) practicable tests are known. Toillustrate this last point, we construct a test for the Burr Type XIIdistribution for which, to our knowledge, not a single test is known aside fromthe classical universal procedures.

Fixed point characterizations of continuous univariate probability distributions and their applications