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# Almost sure testability of classes of densities

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## Abstract

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 allow one 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.

## Suggested Citation

• Luc Devroye & Gábor Lugosi, 1999. "Almost sure testability of classes of densities," Economics Working Papers 375, Department of Economics and Business, Universitat Pompeu Fabra.
• Handle: RePEc:upf:upfgen:375
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File URL: https://econ-papers.upf.edu/papers/375.pdf
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## References listed on IDEAS

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1. Luc Devroye & Gábor Lugosi & Frederic Udina, 1998. "Inequalities for a new data-based method for selecting nonparametric density estimates," Economics Working Papers 281, Department of Economics and Business, Universitat Pompeu Fabra.
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### Keywords

Density estimation; kernel estimate; convergence; testing; asymptotic optimality; minimax rate; minimum distance estimation; total boundedness;

### JEL classification:

• C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

### NEP fields

This paper has been announced in the following NEP Reports:

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