cannot be defined pointwise and exists only as the solution, f in (2) to the inverse problem in the space of generalized functions. When considered in the space of generalized functions the estimators, ̂ f, are viewed as random continuous linear functionals on spaces of well- behaved functions where convergence to generalized derivatives of distribu- tion functions (solutions to the inverse problem) can be established without any assumptions on the underlying distribution. Moreover, convergence of kernel estimators can be faster and even at parametric rates. This result has features common to other results on convergence of random functionals of density as discussed, e.g. in Anderson et al (2012) and is derived here in section 3. This result relies on the derivation of the rate of bias in generalized functions that was provided in Zinde-Walsh (2008) but gives the derivation of the covariance functional that corrects the one in that paper. Conditioning is somewhat awkward and there are many different ways to streamline the representation of conditional measures and distribution func- tions (Chiang and Pollard, 1997, Pfanzagl, 1979 among others). Here we focus on the distribution function F ( x, y ) on R d$_{x}$ × R d$_{y}$ and distribution of y ∈ R d$_{y}$ conditional on x ∈ R $^{d$_{x}$}$. In this case typically the conditional distribution F$_{y}$$_{|}$$_{x}$ function is represented via a fraction ∂ d $^{x}$F ( x,y ) f$_{x}$ ( x ) , where the differentiation operator is applied to the x argument of F ( x, y ) and f$_{x}$ ( x ) represents the density of the marginal distribution. Of course such a repre- sentation makes stringent requirements on the smoothness of the appropriate functions. Here the case of an arbitrary continuous conditioning distribution 7