Maximum likelihood estimation of a multi‐dimensional log‐concave density

By Madeleine Cule, Richard J. Samworth, M. Stewart. 2008

211 citations recorded by the source corpus.

Summary. Let X1,…,Xn be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log‐concave maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non‐constructive, we can reformulate the issue of computing in terms of a non‐differentiable convex optimization problem, and thus combine techniques of computational geometry with Shor's r‐algorithm to produce a sequence that converges to . An R version of the algorithm is available in the package LogConcDEAD—log‐concave density estimation in arbitrary dimensions. We demonstrate that the estimator has attractive theoretical properties both when the true density is log‐concave and when this model is misspecified. For the moderate or large sample sizes in our simulations, is shown to have smaller mean integrated squared error compared with kernel‐based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the expectation–maximization algorithm to fit finite mixtures of log‐concave densities.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b78ecd9592a11b52f57177c5838925891ac74411

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Citations in the corpus

Faithful Variable Screening for High-Dimensional Convex Regression (2014, background): “Cule, Samworth and Stewart [7] and Kim and Samworth [14] analyze log-concave density estimation and prove consistency of the MLE; the latter further show that log-concave density estimation has minimax risk lower bounded by n−2/(d+1) for d ≥ 2, refuting a common notion that the condition of convexit”
Global rates of convergence in log-concave density estimation (2014, background): “Very recently, Doss and Wellner (2013) introduced a new bracketing argument to obtain a rate of convergence of Op(n ) in squared Hellinger distance in the case d = 1, again for a fixed true log-concave density f0.”
Multivariate convex regression with adaptive partitioning (2011, methodology): “, such as production curves, preference functions and value-to-go functions. In statistics, shape restrictions like log-concavity are useful in density estimation, including the multivariate setting (Cule et al. 2010, Cule & Samworth 2010, Schuhmacher & Dum bgen 2010). Although convex regres”
Generalized additive and index models with shape constraints (2014, background): “…certain other shape-constrained estimation problems where boundary effects are also observed, such as log-concave density estimation, it is possible to construct a fully automatic smoothed estimate to alleviate these issues (Dümbgen and Rufibach, 2009; Cule et al., 2010; Chen and Samworth, 2013).”
Selective Classification Can Magnify Disparities Across Groups (2020, background): “Cule et al. (2010) show the mixture is no longer log-concave.”
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.”
Variable selection with error control: another look at stability selection (2011, background): “Log-concavity is a shape constraint that has received a large amount of attention recently (e.g. Walther (2002), Dümbgen and Rufibach (2009) and Cule et al. (2010)), and at first sight it seems reasonable in our context, because, if the summands in expression (4) were independent, then we would…”