In this paper we study the problem of pointwise density estimation from observations with multiplicative measurement errors. We elucidate the main feature of this problem: the influence of the estimation point on the estimation accuracy. In particular, we show that, depending on whether this point is separated away from zero or not, there are two different regimes in terms of the rates of convergence of the minimax risk. In both regimes we develop kernel-type density estimators and prove upper bounds on their maximal risk over suitable nonparametric classes of densities. We show that the proposed estimators are rate-optimal by establishing matching lower bounds on the minimax risk. Finally we test our estimation procedures on simulated data.
|Number of pages||2|
|Journal||Annales de l'institut Henri Poincare (B) Probability and Statistics|
|State||Published - 2020|
Bibliographical noteFunding Information:
This article was prepared within the framework of the HSE University Basic Research Program and funded by the Russian Academic Excellence Project 5-100. Results of Sections 4 and 5 have been obtained under support of the RSF grant No. 18-11-00132. The first author gratefully acknowledges the financial support by the German Research Foundation (DFG) through the Collaborative Research Center 832. The second author gratefully acknowledges the financial support by the Israel Science Foundation (ISF) research grant 361/15. The authors are grateful to an anonymous referee for careful reading and insightful comments that lead to substantial improvements in the paper.
© 2020 Association des Publications de l'Institut Henri Poincaré.
- Density estimation
- Multiplicative censoring
- Multiplicative measurement errors
- Scale mixtures
- The Mellin transform
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty