Abstract. Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein (2009), introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting Bayes rule. The second, motivated by Jiang and Zhang (2009), proposes a new approach to computing the Kiefer-Wolfowitz nonparametric maximum likelihood estimator for mixtures. In contrast to prior methods for these problems, our new approaches are cast as convex optimization problems that can be efficiently solved by modern interior point methods. In particular, we show that the reformulation of the Kiefer-Wolfowitz estimator as a convex optimization problem reduces the computational effort by several orders of magnitude for typical problems, by comparison to prior EM-algorithm based methods, and thus greatly expands the practical applicability of the resulting methods. Our new procedures are compared with several existing empirical Bayes methods in simulations employing the well-established design of Johnstone and Silverman (2004). Some further comparisons are made based on prediction of baseball batting averages. A Bernoulli mixture application is briefly considered in the penultimate section.

The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development

Abstract. For a general definition of depth in data analysis a differential-like calculus is constructed in which the location case (the framework of Tukey's median) plays a fundamental role similar to that of linear functions in the mathematical analysis. As an application, a lower bound for maximal regression depth is proved in the general multidimensional case-as conjectured by Rousseeuw and Hubert, and others. This lower bound is demonstrated to have an impact on the breakdown point of the maximum depth estimator.

Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.Comment: Published in at http://dx.doi.org/10.1214/10-AOS814 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. Hansen, Kooperberg, and Sardy (1998) introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modeling of bivariate densities, regression and hazard functions. These triograms enjoy a natural affine equivariance that offers distinct advantages over competing tensor product methods that are more commonly used in statistical applications.Triograms employ basis functions consisting of linear "tent functions" defined with respect to a triangulation of a given planar domain. As in knot selection for univariate splines, Hansen, et al adopt the regression spline approach of Stone (1994). Vertices of the triangulation are introduced or removed sequentially in an effort to balance fidelity to the data and parsimony.In this paper we explore a smoothing spline variant of the triogram model based on a roughness penalty adapted to the piecewise linear structure of the triogram model. We show that the proposed roughness penalty may be interpreted as a total variation penalty on the gradient of the fitted function. The methods are illustrated with two artificial examples and with an application to estimated quantile surfaces of land value in the Chicago metropolitan area.