Adaptation in multivariate logconcave density estimation
Abstract
We study the adaptation properties of the multivariate logconcave maximum likelihood estimator over three subclasses of logconcave densities. The first consists of densities with polyhedral support whose logarithms are piecewise 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 logconcave density with $d \in \{2,3\}$, we prove a sharp oracle inequality, which in particular implies that the KullbackLeibler risk of the logconcave 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 logconcave maximum likelihood estimator attains the rate $n^{4/7}$ when $d=3$, which is faster than the worstcase rate of $n^{1/2}$. Finally, our third type of subclass consists of densities whose contours are wellseparated; these new classes are constructed to be affine invariant and turn out to contain a wide variety of densities, including those that satisfy Hölder regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the logconcave maximum likelihood estimator attains a risk bound of order $n^{\min\bigl(\frac{\beta+3}{\beta+7},\frac{4}{7}\bigr)}$ when $d=3$ over the class of $\beta$Hölder logconcave densities with $\beta\in (1,3]$, again up to a polylogarithmic factor.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.11634
 Bibcode:
 2018arXiv181211634F
 Keywords:

 Mathematics  Statistics Theory;
 62G07;
 62G20
 EPrint:
 97 pages, 6 figures