We are now going to apply our version of Kullback’s theorem based in two hypothesis to the non-parametric regression model. Assume first the following conditions:

In the last publication, we defined a minimax lower bound as $latex \displaystyle \mathcal{R}^{*}\geq cs_{n} &fg=000000$ where $latex {\mathcal{R}^{*}\triangleq\inf_{\hat{f}}\sup_{f\in\mathcal{F}}\mathbb E\left[d^{2}(\hat{f}_{n},f)\right]}&fg=000000$ and $latex {s_{n}\rightarrow0}&fg=000000$. The big issue with this definition is to take the supremum over a massive set $latex {\mathcal{F}}&fg=000000$ and then the infimum over all the possible estimators of $latex {f}&fg=000000$.