Theorem 1. Under Assumption 1 for each of the models 1-4 the corre- sponding convolution equations of Table 1 hold in the generalized functions space S $^{∗}$. The proof is in the derivations of the following subsection. Assumption 1 requires considering all the functions defined by the model as elements of the space S $^{∗}$, but if the functions (e.g. densities, the con- ditional moments) exist as regular functions, the convolutions are just the usual convolutions of functions, on the other hand, the assumption allows to consider convolutions for cases where distributions are not absolutely contin- uous. The case of the classical measurement error is well known in the literature. The concept of error independent of the variable of interest is applicable to many problems in seismology, image processing, where it may be assumed that the source of the error is unrelated to the signal. In e.g. Cunha et al. (2010) it is assumed that some constructed measurement of ability of a child derived from test scores fits into this framework. As is well-known in regression a measurement error in the regressor can result in a biased estimator (attenuation bias). 2.2.2 Measurement error models and derivation of the correspond- ing equations. 1. The classical measurement error model. 15