## 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 …

## Minimax Lower Bounds using the Total Variation Divergence

Remember that we have supposed two hypothesis $latex {\left\{ f_{0},f_{1}\right\} }&fg=000000$ elements of $latex {\mathcal{F}}&fg=000000$. Denote $latex {P_{0}}&fg=000000$ and $latex {P_{1}}&fg=000000$ two probability measures under $latex {(\mathcal{X},\mathcal{A})}&fg=000000$ under $latex {f_{0}}&fg=000000$ and $latex {f_{1}}&fg=000000$ respectively. If $latex {P_{0}}&fg=000000$ and $latex {P_{1}}&fg=000000$ are very “close”, then it is hard to distinguish $latex {f_{0}}&fg=000000$ and $latex {f_{1}}&fg=000000$ and …