in support of density ˆ f$_{x}$ ( x ) = h d − $^{1}$b + o$_{p}$ ( h d − 1 ) with some b > 0 and d = ln 2 ln 3 < 1. In the univariate case this is discussed in Example 5 in Zinde-Walsh (2008), where for the Cantor distribution it is noted that though ˆ f$_{x}$ ( x ) may diverge, h 1 − d ˆ f$_{x}$ ( x ) is bounded and bounded away from zero. Then, even though the limit density does not exist by rescaling it is possible to establish the convergence rate of the estimator of the conditional distribution as a functional on D$_{0}$ ( R d$_{x}$ ) ; the rate is n − 1 2 h 1 − d and is faster than the root-n rate. Consider now a conditional moment of a function g ( y ) , of y ∈ R d$_{y}$ : E$_{y}$$_{|}$$_{x}$g ( y ) = m ( x ) , with m ( x ) measurable with respect to F$_{x}$. When the conditional density function exists in L$_{1}$ we write m ( x ) = ∫ g ( y ) f$_{y}$$_{|}$$_{x}$ ( x, y ) dy (assuming that the integral exists). As a generalized func- tion (in x ) m ( x ) can be presented on the space D ( W ) ; W = (0 , 1) d$_{x}$ by the value of the functional for ψ : To give meaning to ( m, ψ ) regardless of the existence of the conditional density as a function, ∫ g ( y ) f$_{y}$$_{|}$$_{x}$ ( x, y ) dy needs to be characterized as a gener- alized function on D ( W ) . To make this possible for an arbitrary distribution ( m, ψ ) = ∫ m ( x ) ψ ( F ( x )) dF ( x ) = ∫ [∫ g ( y ) f$_{y}$$_{|}$$_{x}$ ( x, y ) dy ] ψ ( F ( x )) dF ( x ) . 25 6 Conditional moments