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

✍ Scribed by Sam Efromovich

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 172 KB
• Volume: 63
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(03)00087-7

✦ Synopsis

The fact that signal-in-noise ÿltering 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 1 2 . 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 di erentiable. 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 di erentiable, it is always possible to ÿnd a heteroscedastic regression model that is nonequivalent to the corresponding ÿltering model. As a result, the statistician should be vigilant in using ÿltering models for the analysis of heteroscedastic regressions.