Assumption 2 is applied to model 1 for a = f$_{x}$ ∗ , b = f$_{u}$ ; to model 2 with a = f$_{z}$, b = f$_{u}$ ; to model 3 with a = f$_{x}$ ∗ , b = f$_{u}$ and with a = h$_{k}$, b = f$_{u}$, for all k = 1 , ..., d ; to model 4a for a = f$_{x}$ ∗ , or f$_{u}$$_{x}$, or h$_{k}$ for all k and b = f$_{u}$ ; to model 5 with a = f$_{x}$ ∗ , or gf$_{x}$ ∗ , or h$_{k}$f$_{x}$ ∗ and b = f$_{u}$ ; to model 6 with a = f$_{z}$, or g and b = f$_{u}$ ; to model 7 with a = g or h$_{k}$ and b = f$_{u}$. Assumption 2 is a high-level assumption that is a sufficient condition for a solution to the models 1-4 and 6-7 to exist. Some additional conditions are needed for model 5 and are provided below. Assumption 2 is automatically satisfied for generalized density functions, so is not needed for models 1 and 2. Denote by ¯ D ⊂ S ∗ the subset of general- ized derivatives of distribution functions (corresponding to Borel probability measures in R $^{d}$) then in models 1 and 2 A = B = ¯ D ; and for the character- istic functions there are correspondingly no restrictions; denote the set of all characteristic functions, F t ( ¯ D ) ⊂ S $^{∗}$, by ¯ C. Below a (non-exhaustive) list of nonparametric classes of generalized func- tions that provide sufficient conditions for existence of solutions to the models here is given. The classes are such that they provide minimal or often no restrictions on one of the functions and restrict the class of the other in order that the assumptions be satisfied. In models 3 and 4 the functions h$_{k}$ are transformed into derivatives of continuous characteristic functions. An assumption that either the charac- teristic function of x ∗ or the characteristic function of u be continuously differentiable is sufficient, without any restrictions on the other to ensure 32