On the limit in the equivalence between heteroscedastic regression and filtering model

The fact that signal-in-noise filtering model can serve as a prototype for nonparametric regression is well known and it is widely used in the nonparametric curve estimation theory. So far the only known examples showing that the equivalence may not hold have been about homoscedastic regressions where regression functions have the smoothness index at most . As a result, the possibility of a nonequivalence and its consequences have been widely ignored in the literature by assuming that an underlying regression function is smooth enough, for instance, it is differentiable. This note presents another interesting example of a nonequivalence. It is shown that, regardless of how many times the regression function is assumed to be differentiable, it is always possible to find a heteroscedastic regression model that is nonequivalent to the corresponding filtering model. As a result, the statistician should be vigilant in using filtering models for the analysis of heteroscedastic regressions.