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Adaptive log-density estimation

Authors
Bak, Kwan-YoungKoo, Ja-Yong
Issue Date
6월-2020
Publisher
SPRINGER HEIDELBERG
Keywords
l(1) penalty; Log-density estimation; Minimax adaptivity; Model selection consistency; Oracle inequality; Wavelet basis
Citation
JOURNAL OF THE KOREAN STATISTICAL SOCIETY, v.49, no.2, pp.293 - 323
Indexed
SCIE
SCOPUS
KCI
Journal Title
JOURNAL OF THE KOREAN STATISTICAL SOCIETY
Volume
49
Number
2
Start Page
293
End Page
323
URI
https://scholar.korea.ac.kr/handle/2021.sw.korea/55453
DOI
10.1007/s42952-019-00018-8
ISSN
1226-3192
Abstract
This study examines an adaptive log-density estimation method with an 1-type penalty. The proposed estimator is guaranteed to be a valid density in the sense that it is positive and integrates to one. The smoothness of the estimator is controlled in a data-adaptive way via 1 penalization. The advantages of the penalized log-density estimator are discussedwith an emphasis onwavelet estimators. Theoretical properties of the estimator are studied when the quality of fit is measured by theKullback-Leibler divergence (relative entropy). A nonasymptotic oracle inequality is obtained assuming a near orthogonality condition on the given dictionary. Based on the oracle inequality, selection consistency and minimax adaptivity are proved under some regularity conditions. The proposed method is implemented with a coordinate descent algorithm. Numerical illustrations based on the periodized Meyer wavelets are performed to demonstrate the finite sample performance of the proposed estimator.
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