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Estimation of a unimodal distribution function. (English) Zbl 0611.62037
Author’s summary: This paper deals with the problem of efficiently estimating (asymptotically minimax) a distribution function when essentially nothing is known about it except that it is unimodal.
The sample distribution function \(F_ n\) is shown to be asymptotically minimax among the family \({\mathcal E}\) of all unimodal distribution functions. Since \(F_ n\) does not belong to this family, estimators belonging to this family are constructed and are shown to be asymptotically minimax relative to the collection of subfamilies of \({\mathcal E}\).
Reviewer: Zhao Lincheng

MSC:
62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions
62C20 Minimax procedures in statistical decision theory
62G20 Asymptotic properties of nonparametric inference
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