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 […]

A first minimax lower bound in the two hypothesis scenario

Photos of  Johann Radon and Otto Nikodym. Sources: Apprendre les Mathématiques and Wikipedia. Consider the simplest case, $latex {M=1}&fg=000000$ with two hypothesis $latex {\{f_{1},f_{2}\}}&fg=000000$ belonging to $latex {\mathcal{F}}&fg=000000$. According to the last post, we need only to find lower bounds for the minimax probability of error $latex {p_{e,1}}&fg=000000$. Today, we will find a bound using […]

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 $latex {f}&fg=000000$.

The Delta method: Main Result

Let $latex {T_{n}}&fg=000000$ an estimator of $latex {\theta}&fg=000000$, we want to estimate the parameter $latex {\phi(\theta)}&fg=000000$ where $latex {\phi}&fg=000000$ is a known function. It is natural to estimate $latex {\phi(\theta)}&fg=000000$ by $latex {\phi(T_{n})}&fg=000000$. Now, we can then ask: How the asymptotic properties of $latex {T_{n}}&fg=000000$ could be transfer to $latex {\phi(T_{n})}&fg=000000$?

Mathematics book suggestions

A group of colleagues from the Faculty of Mathematics of the University of Costa Rica, are collecting suggestions for modern books in mathematics to buy some for their lectures and research. The list is in and we will appreciate any recommendation that you could give us in the comments. Thank you.