Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density

By Madeleine Cule, Richard J. Samworth. 2009

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We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in R d . Our study covers both the case where the true underlying density is log-concave, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, con- vergence in distribution implies much stronger types of convergence - in particular, it implies convergence in Hellinger distance and even in cer- tain exponentially weighted total variation norms. In our main result, we prove the existence and uniqueness of a log-concave density that minimises the Kullback-Leibler divergence from the true density over the class of all log-concave densities, and also show that the log-concave maximum like- lihood estimator converges almost surely in these exponentially weighted total variation norms to this minimiser. In the case of a correctly specified model, this demonstrates a strong type of consistency for the estimator; in a misspecified model, it shows that the estimator converges to the log- concave density that is closest in the Kullback-Leibler sense to the true density.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f81697741fd1614e0a74e2ba9d376009d115fef0

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Citations in the corpus

Semiconvex Regression for Metamodeling-Based Optimization (2014, methodology): “Logconcave functions are a popular choice for density estimation [23, 20, 21, 60], but have potential to fit response surfaces as well because they are strongly unimodal.”
Recent Progress in Log-Concave Density Estimation (2017, background): “The constraint of log-concavity rules out certain pathologies and means we can strengthen certain convergence statements: Theorem 7.1 (Cule and Samworth (2010); Schuhmacher, Hüsler and Dümbgen (2011)).”
Log-Concavity and Strong Log-Concavity: a review (2014, background): “…much recent work on nonparametric estimation for the class of log-concave distributions on R and Rd: for example, see Dümbgen and Rufibach (2009), Cule and Samworth (2010), Cule, Samworth and Stewart (2010), Walther (2009), Balabdaoui, Rufibach and Wellner (2009), and Henningsson and Astrom…”
Global rates of convergence in log-concave density estimation (2014, background): “On the theoretical side, through a series of papers (Pal, Woodroofe, and Meyer, 2007; Dümbgen and Rufibach, 2009; Seregin and Wellner, 2010; Schuhmacher and Dümbgen, 2010; Cule and Samworth, 2010; Dümbgen et al., 2011), we now have a fairly complete understanding of the global consistency properties”
APPROXIMATION BY LOG-CONCAVE DISTRIBUTIONS, WITH APPLICATIONS TO REGRESSION (2010, background): “To the best of our knowledge, their results are not directly applicable in the setting of log-concave densities, which is treated by Cule and Samworth (2010).”
Smoothed log-concave maximum likelihood estimation with applications (2011, methodology): “15 of Dümbgen, Samworth and Schuhmacher (2011) and another application of Proposition 2(c) of Cule and Samworth (2010) that there exists a > 0 such that ∫”
Maximum likelihood estimation of a multi‐dimensional log‐concave density (2008, background): “It is also proved in Cule and Samworth (2010), lemma 1, that, given any log-concave density fÅ, we can always find a0 >0 and b0 ∈R such that fÅ.x/ exp.−a0‖x‖+b0/, so there is no danger of the conclusion being vacuous.”
Handbook of Big Data (2016, background): “Recent statistical research on the topic attempts to determine tight upper and lower bounds on the minimax rate of convergence [17,18] in the context of the MLE.”
Multivariate convex regression with adaptive partitioning (2011, methodology): “Key words: Nonparametric regression, shape constraint, convex regression, treed linear model, adaptive partitioning”
QUASI-CONCAVE DENSITY ESTIMATION (2010, background): “For maximum likelihood estimators in Rd , Cule and Samworth (2010) establish consistency for estimators of a log-concave density, and Seregin and Wellner (2009) for estimators of convex-transformed densities.”