We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log-supermodular (MTP₂) distributions and log-L^♮-concave (LLC) distributions. In both cases we also assume log-concavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n≥3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0,1}^d or in ${ℝ}^2$ under MTP₂, and for samples in ${\mathbb{Q}}^d$ under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.

中文翻译：

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完全正对数凹面密度的最大似然估计

我们研究了两类多元分布的非参数最大似然估计，这些分布意味着强的正相关形式；即对数超模 ( MTP ₂ ) 分布和对数- L ^♮ -凹( LLC ) 分布。在这两种情况下，我们还假设对数凹度以确保似然函数的有界性。给定来自我们的分布之一的n 个独立且同分布的随机向量，当n ≥ 3时，最大似然估计量 (MLE) 以概率 1 的形式存在并且是唯一的 ae 。这与环境维度d无关. 我们推测 MLE 始终是帐篷函数的指数。我们在 {0,1} ^d或${ℝ}^2$在 MTP _{2 下}，对于样品${\mathbb{Q}}^d$在有限责任公司下。最后，我们提供了一种用于计算最大似然估计的条件梯度算法。