On Information Pooling, Adaptability and Superefficiency in Nonparametric Function Estimation
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Bayes rules
information pooling
minimax
minimum risk inequalities
nonparametric regression
orthogonal series
separable rules
superefficiency
wavelets
white noise
Applied Mathematics
Statistics and Probability
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Abstract
The connections between information pooling and adaptability as well as superefficiency are considered. Separable rules, which figure prominently in wavelet and other orthogonal series methods, are shown to lack adaptability; they are necessarily not rate-adaptive. A sharp lower bound on the cost of adaptation for separable rules is obtained. We show that adaptability is achieved through information pooling. A tight lower bound on the amount of information pooling required for achieving rate-optimal adaptation is given. Furthermore, in a sharp contrast to the separable rules, it is shown that adaptive non-separable estimators can be superefficient at every point in the parameter spaces. The results demonstrate that information pooling is the key to increasing estimation precision as well as achieving adaptability and even superefficiency.