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Almost sure testability of classes of densities
Devroye, Luc; Lugosi, Gábor
Universitat Pompeu Fabra. Departament d'Economia i Empresa
Let a class $\F$ of densities be given. We draw an i.i.d.\ sample from a density $f$ which may or may not be in $\F$. After every $n$, one must make a guess whether $f \in \F$ or not. A class is almost surely testable if there exists such a testing sequence such that for any $f$, we make finitely many errors almost surely. In this paper, several results are given that allowone to decide whether a class is almost surely testable. For example, continuity and square integrability are not testable, but unimodality, log-concavity, and boundedness by a given constant are.
Statistics, Econometrics and Quantitative Methods
density estimation
kernel estimate
asymptotic optimality
minimax rate
minimum distance estimation
total boundedness
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