Density estimation is a fundamental statistical problem. Many methods are either sensitive to model misspecification (parametric models) or difficult to calibrate, especially for multivariate data (nonparametric smoothing methods). We propose an alternative approach using maximum likelihood under a qualitative assumption on the shape of the density, specifically log-concavity. The class of log-concave densities includes many common parametric families and has desirable properties. For univariate data, these estimators are relatively well understood, and are gaining in popularity in theory and practice. We discuss extensions for multivariate data, which require different techniques.

After establishing existence and uniqueness of the log-concave maximum likelihood estimator for multivariate data, we see that a reformulation allows us to compute it using standard convex optimization techniques. Unlike kernel density estimation, or other nonparametric smoothing methods, this is a fully automatic procedure, and no additional tuning parameters are required.

Since the assumption of log-concavity is non-trivial, we introduce a method for assessing the suitability of this shape constraint and apply it to several simulated datasets and one real dataset. Density estimation is often one stage in a more complicated statistical procedure. With this in mind, we show how the estimator may be used for plug-in estimation of statistical functionals. A second important extension is the use of log-concave components in mixture models. We illustrate how we may use an EM-style algorithm to fit mixture models where the number of components is known. Applications to visualization and classification are presented. In the latter case, improvement over a Gaussian mixture model is demonstrated.

Performance for density estimation is evaluated in two ways. Firstly, we consider Hellinger convergence (the usual metric of theoretical convergence results for nonparametric maximum likelihood estimators). We prove consistency with respect to this metric and heuristically discuss rates of convergence and model misspecification, supported by empirical investigation. Secondly, we use the mean integrated squared error to demonstrate favourable performance compared with kernel density estimates using a variety of bandwidth selectors, including sophisticated adaptive methods.

Throughout, we emphasise the development of stable numerical procedures able to handle the additional complexity of multivariate data.