GLOBAL RATES OF CONVERGENCE IN LOG-CONCAVE DENSITY ESTIMATION

The estimation of a log-concave density on Rd represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size n can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk smaller than order n−4/5, when d=1, and order n−2/(d+1) when d≥2. In particular, this reveals a sense in which, when d≥3, log-concave density estimation is fundamentally more challenging than the estimation of a density with two bounded derivatives (a problem to which it has been compared). Second, we show that for d≤3, the Hellinger ε-bracketing entropy of a class of log-concave densities with small mean and covariance matrix close to the identity grows like max{ε−d/2,ε−(d−1)} (up to a logarithmic factor when d=2). This enables us to prove that when d≤3 the log-concave maximum likelihood estimator achieves the minimax optimal rate (up to logarithmic factors when d=2,3) with respect to squared Hellinger loss.The research of Richard J. Samworth was supported by an EPSRC Early Career Fellowship and a grant from the Leverhulme Trust

ADAPTATION IN LOG-CONCAVE DENSITY ESTIMATION

The log-concave maximum likelihood estimator of a density on the real line based on a sample of size $n$ is known to attain the minimax optimal rate of convergence of $O(n^{-4/5})$ with respect to, e.g., squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is $k$-affine (i.e.\ made up of $k$ affine pieces), provided $k$ is not too large. Our results use two different techniques: the first relies on a new Marshall's inequality for log-concave density estimation, and reveals that when the true density is close to log-linear on its support, the log-concave maximum likelihood estimator can achieve the parametric rate of convergence in total variation distance. Our second approach depends on local bracketing entropy methods, and allows us to prove a sharp oracle inequality, which implies in particular that the rate of convergence with respect to various global loss functions, including Kullback--Leibler divergence, is $O\bigl(\frac{k}{n}\log^{5/4} n\bigr)$ when the true density is log-concave and its logarithm is close to $k$-affine.AKH Kim: National Research Foundation of Korea (NRF) grant 2017R1C1B5017344. A Guntuboyina: NSF Grant DMS-1309356. RJ Samworth: EPSRC Early Career Fellowship and a grant from the Leverhulme Trust

GLOBAL RATES OF CONVERGENCE IN LOG-CONCAVE DENSITY ESTIMATION

The estimation of a log-concave density on Rd represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size n can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk smaller than order n−4/5, when d=1, and order n−2/(d+1) when d≥2. In particular, this reveals a sense in which, when d≥3, log-concave density estimation is fundamentally more challenging than the estimation of a density with two bounded derivatives (a problem to which it has been compared). Second, we show that for d≤3, the Hellinger ε-bracketing entropy of a class of log-concave densities with small mean and covariance matrix close to the identity grows like max{ε−d/2,ε−(d−1)} (up to a logarithmic factor when d=2). This enables us to prove that when d≤3 the log-concave maximum likelihood estimator achieves the minimax optimal rate (up to logarithmic factors when d=2,3) with respect to squared Hellinger loss

ADAPTATION IN MULTIVARIATE LOG-CONCAVE DENSITY ESTIMATION

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 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 log-concave density with $d \in \{2,3\}$, 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 H\"older 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 $n^{-\min\bigl(\frac{\beta+3}{\beta+7},\frac{4}{7}\bigr)}$ when $d=3$ over the class of $\beta$-H\"older log-concave densities with $\beta\in (1,3]$, again up to a polylogarithmic factor.EPSRC Fellowship EP/P031447/1 Leverhulme Trust grant RG8176