Chapter 5: Minmax Lower Bounds

Both questions ask about some form of optimality. The first one is about optimality of an estimator, whereas the second one is about optimality of a bound. The difficulty of these questions varies depending on whether we are looking for a positive or a negative answer. Indeed, a positive answer to these questions simply consists in finding a better proof for the estimator we have studied (question 1.) or simply finding a better estimator, together with a proof that it performs better (question 2.). A negative answer is much more arduous. For example, in question 2., it is a statement about all estimators. How can this be done? The answer lies in information theory (see [CT06] for a nice introduction). In this chapter, we will see how to give a negative answer to question 2. It will imply a negative answer to question 1.