Abstract

We present a new approach for inference about a univariate log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf F. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus, overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for F which we introduce here has attractive computational and statistical properties: It allows to bring modern tools from optimization to bear on this problem via difference of convex programming, and it results in optimal statistical inference. We show that the width of the resulting confidence bands converges at nearly the parametric $n^{-\frac{1}{2}}$ rate when the log density is k-affine. Supplementary materials for this article are available online.

摘要

我们提出了一种推断单变量对数凹分布的新方法：我们建议将对数凹约束合并到 cdf F的适当非参数置信集中，而不是使用最大似然法。这种方法的优点是它自动提供统计不确定性的度量，因此克服了最大似然估计的显着限制。特别是，我们展示了如何为具有有限样本保证置信水平的密度构建置信带。F的非参数置信集我们在这里介绍的具有吸引人的计算和统计特性：它允许通过凸规划的差异将现代工具从优化中引入到这个问题上，并产生最佳统计推断。我们表明，所得置信带的宽度几乎收敛于参数$n^{-\frac{\mathrm{1个}}{\mathrm{2个}}}$对数密度为k仿射时的速率。本文的补充材料可在线获取。