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ADAPTATION IN MULTIVARIATE LOG-CONCAVE DENSITY ESTIMATION

Authors
Feng, Oliver Y.Guntuboyina, AdityanandKim, Arlene K. H.Samworth, Richard J.
Issue Date
2월-2021
Publisher
INST MATHEMATICAL STATISTICS-IMS
Keywords
Multivariate adaptation; bracketing entropy; log-concavity; contour separation; maximum likelihood estimation
Citation
ANNALS OF STATISTICS, v.49, no.1, pp.129 - 153
Indexed
SCIE
SCOPUS
Journal Title
ANNALS OF STATISTICS
Volume
49
Number
1
Start Page
129
End Page
153
URI
https://scholar.korea.ac.kr/handle/2021.sw.korea/49649
DOI
10.1214/20-AOS1950
ISSN
0090-5364
Abstract
We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over three subclasses of log-concave densities. The first consists of densities with polyhedral support whose logarithms are piece-wise affine. The complexity of such densities f can be measured in terms of the sum Gamma(f) of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by f. Given n independent observations from a d-dimensional log-concave density with d is an element of {2, 3}, we prove a sharp oracle inequality, which in particular implies that the Kullback-Leibler risk of the log-concave maximum likelihood estimator for such densities is bounded above by Gamma(f)/n., up to a polylogarithmic factor. Thus, the rate can be essentially parametric, even in this multivariate setting. For the second type of adaptation, we consider densities that are bounded away from zero on a polytopal support; we show that up to polylogarithmic factors, the log-concave maximum likelihood estimator attains the rate n(-4/7) when d = 3, which is faster than the worst-case rate of n(-1/2). Finally, our third type of subclass consists of densities whose contours are well separated; these new classes are constructed to be affine invariant and turn out to contain a wide variety of densities, including those that satisfy Holder regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the log-concave maximum likelihood estimator attains a risk bound of order n(-min()(beta+3/)(beta+7, 4/7)) when d = 3 over the class of beta-Holder log-concave densities with beta > 1, again up to a polylogarithmic factor.
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