Locally adaptive confidence bands
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Date
2016
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Abstract
We develop honest and locally adaptive confidence bands for probability
densities. They provide substantially improved confidence statements in
case of inhomogeneous smoothness, and are easily implemented and visualized.
The article contributes conceptual work on locally adaptive inference
as a straightforward modification of the global setting imposes severe obstacles
for statistical purposes. Among others, we introduce a statistical notion
of local Hölder regularity and prove a correspondingly strong version of local
adaptivity. We substantially relax the straightforward localization of
the self-similarity condition in order not to rule out prototypical densities.
The set of densities permanently excluded from the consideration is shown
to be pathological in a mathematically rigorous sense. On a technical level,
the crucial component for the verification of honesty is the identification
of an asymptotically least favorable stationary case by means of Slepian's
comparison inequality.