Example with two hypothesis: Regression case

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:

The Kullback’s version for the minimax lower bound with two hypothesis

Photos of (left to right) Solomon Kullback, Richard A. Leibler and Lucien Le Cam. Sources: NSA Cryptologic Hall of Honor (1, 2) and MacTutor. We saw the last time how to find lower bounds using the total variation divergence.  Even so, conditions with the Kullback-Leiber divergence are easier to verify than the total divergence and …

A general reduction scheme for minimax lower bounds

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 $latex {f}&fg=000000$.