Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on $\mathbb{R}^d$ that are log-concave, and we study geometric properties of the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e. the face poset of their secondary polytope. We prove that this map is surjective. In fact, every regular subdivision arises in the MLE for some set of weights with positive probability, but coarser subdivisions appear to be more likely to arise than finer ones. To quantify these results, we introduce a continuous version of the secondary polytope, whose dual we name the Samworth body. This article establishes a new link between geometric combinatorics and nonparametric statistics, and it suggests numerous open problems.
翻译:形状限制的密度估计是数学统计中的一个重要专题。 我们侧重于对数集合值的 $\ mathbb{R ⁇ d$的密度, 我们研究加权样品的最大概率估计值的几何特性。 Cule、 Samworth 和 Stewart 显示, 最佳对数计算密度的对数是线性的, 并且支持在样本的常规分层中。 这定义了从重量空间到样本常规分层的地图, 即其二级多功能体的面部形状。 我们证明这幅地图是推测性的。 事实上, 每一个常规的分层在 mLE 中出现一些正概率的重量, 但粗略的次视图似乎比细微的对数更可能出现。 为了量化这些结果, 我们引入了一个二级多功能的连续版本, 其二元体由我们命名为 Samworth 体。 这篇文章在几何测量的合成和不精确的统计中建立了新的链接。