Shape constraints yield flexible middle grounds between fully nonparametric and fully parametric approaches to modeling distributions of data. The specific assumption of log-concavity is motivated by applications across economics, survival modeling, and reliability theory. However, there do not currently exist valid tests for whether the underlying density of given data is log-concave. The recent universal inference methodology provides a valid test. The universal test relies on maximum likelihood estimation (MLE), and efficient methods already exist for finding the log-concave MLE. This yields the first test of log-concavity that is provably valid in finite samples in any dimension, for which we also establish asymptotic consistency results. Empirically, we find that a random projections approach that converts the d-dimensional testing problem into many one-dimensional problems can yield high power, leading to a simple procedure that is statistically and computationally efficient.

翻译：形状约束在数据分布建模中提供了完全非参数方法与完全参数方法之间的灵活折衷。对数据对数凹性的特定假设源于经济学、生存建模和可靠性理论中的应用。然而，目前尚缺乏有效检验给定数据潜在密度是否为对数凹性的方法。近期提出的通用推理方法提供了一种有效的检验手段。该通用检验依赖于最大似然估计（MLE），而已有高效方法可用于求解对数凹性最大似然估计。这首次在任意维度下证明了有限样本中有效的对数凹性检验，并建立了渐近一致性结果。实验表明，采用随机投影方法将d维检验问题转化为多个一维问题，能够实现高检验功效，从而形成一种统计与计算双重高效的简洁流程。