Estimating the mode of a unimodal distribution is a classical problem in statistics. Although there are several approaches for point-estimation of mode in the literature, very little has been explored about the interval-estimation of mode. Our work proposes a collection of novel methods of obtaining finite sample valid confidence set of the mode of a unimodal distribution. We analyze the behaviour of the width of the proposed confidence sets under some regularity assumptions of the density about the mode and show that the width of these confidence sets shrink to zero near optimally. Simply put, we show that it is possible to build finite sample valid confidence sets for the mode that shrink to a singleton as sample size increases. We support the theoretical results by showing the performance of the proposed methods on some synthetic data-sets. We believe that our confidence sets can be improved both in construction and in terms of rate.

翻译：单峰分布众数的估计是统计学中的一个经典问题。尽管文献中存在多种众数点估计方法，但关于众数区间估计的研究却鲜有探索。本文提出了一系列构建单峰分布众数有限样本有效置信集的新方法。在密度函数于众数处满足一定正则性假设的条件下，我们分析了所提置信集宽度的渐近行为，证明这些置信集的宽度能以接近最优的速率收敛至零。简而言之，我们证明了可以构建随着样本量增加而收缩至单点集的有限样本有效众数置信集。通过合成数据集的实验，我们验证了所提方法的性能。我们相信所提出的置信集在构造方法和收敛速率方面均存在改进空间。