and thus the inverse operator is not continuous. Thus although a solution to the inverse problem in the L$_{1}$ space exists for absolutely continuous distributions, the problem is not well-posed. By contrast, in the appropriate space of generalized functions the solution to the density problem exists without any restrictions on the distribution function and is well-posed; as proved in section 2 below this follows from the known properties of generalized functions. The fact that generalized functions can be useful when non-differentiability prevents the use of Taylor expansions was discussed e.g. in Phillips (1991) for LAD estimation, and continued in some econometric literature that followed. The statistical inverse problem is solved often with a kernel density es- timator. Consider a random sample of observations from a distribution F, { x$_{i}$ } n i $_{=1}$, x$_{i}$ ∈ R $^{k}$. With a chosen kernel function, K and bandwidth (vector) h the estimator is where h has components h$_{1}$, ...h$_{k}$ and K ( x$_{i}$ − x h ) is a multivariate function with the argument ( x$_{i}$ − x h ) = (( x$_{i}$$_{1}$ − x 1 h 1 ) , ..., ( x$_{ik}$ − x$_{k}$ h$_{k}$ )) . We shall proceed with the following assumption on the kernel. (a). K ( w ) is an ordinary bounded function on R $^{k}$; ∫ K ( w ) dw = 1; (b). Support of K belongs to [ − 1 , 1] $^{k}$; Assumption 1 (kernel). ̂ f ( x ) = 1 n Π k j $_{=1}$h$_{j}$ n ∑ i =1 K ( x$_{i}$ − x h ) , (3) 5