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 …

###### Monthly Archives: October 2012

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 …

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 $latex {f}&fg=000000$.

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 …

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 …