attenuation bias in regression; also in the convolution equation the unknown function is directly expressed via the known ones when the distribution of Berkson error is known. For discussion see Carrol et al (2006), Meister (2009), and Wang (2004). 

Models 3. and 4. The classical measurement error with another observa- tion. 

In 3., 4. in the classical measurement error model the error distribution is not known but another observation for the mis-measured variable is available; this case has been treated in the literature and is reviewed in Carrol et al (2006), Chen et al (2011). In econometrics such models were examined by Li and Vuong (1998), Li (2002), Schennach (2004) and subsequently others (see e.g. the review by Chen et al, 2011). In case 3 the additional observation contains an error that is not necessarily independent, just has conditional mean zero. 

Note that here the multivariate case is treated where arbitrary depen- dence for the components of vectors is allowed. For example, it may be of interest to consider the vector of not necessarily independent latent abilities or skills as measured by different sections of an IQ test, or the GRE scores. 

Extra measurements provide additional equations. Consider for any k = 1 , ...d the function of observables w$_{k}$ defined by density weighted expecta- tion E ( x$_{k}$f$_{z}$ ( z ) | z ) as a generalized function; it is then determined by the values of the functional ( w$_{k}$, ψ ) for every ψ ∈ S. Note that by assumption E ( x$_{k}$f$_{z}$ ( z ) | z ) = E ( x ∗ $_{k}$f$_{z}$ ( z ) | z ); then for any ψ ∈ S the value of the functional: 

