## 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 $latex {f}&fg=000000$.

## Introduction to Minimax Lower Bounds

In my most recent research, I’m working on finding “Minimax Lower Bounds” for some kind of estimators. Therefore,  to learn a little more and get my ideas clear, I’ll going to start a series of posts about the topic. I pretend to make some review in the general method and introduce some bounds depending on […]

## The Delta Method: Applications

This week I am going to present three applications of the Delta method theorem. The first is a direct one and it is about the behavior in distribution of the sample variance. The second one is an hypothesis test in the variance when the sample is normal. Finally, the third is an interesting application in […]

## 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 $latex {\phi(T_{n})}&fg=000000$?

## Weak Law of Large Numbers and Central Limit Theorem via the Levy’s continuity theorem

The Levy’s continuity theorem is a very important tool in the statistical machinery. For example, it will give us two simple proofs to two classical statistical problems: The Law of Large Numbers and the Central Limit Theorem.

## Revista de Matemática: Teoría y Aplicaciones. Vol 19, No 2 (2012)

The newest edition of the mathematical journal of Costa Rica “Revista de Matemática: Teoría y Aplicaciones. Vol 19, No 2 (2012).” is available here http://bit.ly/PDfXDK

## 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 http://bit.ly/Ojk2Ou and we will appreciate any recommendation that you could give us in the comments. Thank you.