content_image/325347.jpg
not known additional conditions are needed to identify g. If another obser- vation, x, on x ∗ is available: x = x ∗ + u$_{x}$ conditions under which a unique solution exists for the density of interest, g, were given by Kotlyarski (1967) in the case of independence between x ∗ and u$_{x}$ ; the approach there was to consider the joint characteristic function of the two measurements. 

The assumption of independence of the error in the second measurement may be too strong for some applications, for example in Cunha, Heckman and Schennach (2010) one measurement for the latent variable representing a skill of a child was constructed from test scores where one could plausibly assume independence for measurement error, but the extra measurements came from reports by teachers and parents where such an assumption could be unrealistic. Assume that for the error of the second measurement, u$_{x}$, the conditional expectation is zero: E ( u$_{x}$ | x $^{∗}$, u ) = 0 (but heteroscedasticity is not ruled out). Denote by x$_{k}$ the k − th component of x = ( x$_{1}$, ..., x$_{d}$ ) and by h$_{k}$ ( x ) the function x$_{k}$g ( x ), and assume existence of density weighted conditional moments w$_{2}$$_{k}$ ( z ) = E ( w ( z ) x$_{k}$ | z ) . Then in addition to the equation (2) more convolution equations can be written: 

<img src='content_image/325347.jpg'>

Indeed (with integration here and everywhere in this paper over the whole 

