On Adaptability And Information Pooling in Nonparametric Function Estimation

It is well known that it is possible to achieve adaptation for “free” in function estimation under a global loss. It is unclear, however, why and how the adaptability is achieved. In this article we show that adaptability is achieved through information pooling. It is first shown that separable rules, which figure prominently in wavelet and other orthogonal series methods, lack adaptability; they are necessarily not rate-adaptive. A sharp lower bound on the cost of adaptation for separable rules is obtained. We then derive a tight lower bound on the amount of information pooling required for achieving global adaptability. Moreover, in a sharp contrast to the separable rules, it is shown that adaptive nonseparable estimators can be superefficient at every point in the parameter spaces. The results demonstate that information pooling is the key to increase estimation precision and achieve adaptability and even superefficiency.