Finite sample valid confidence sets of mode

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.