Testing for Antimodes

Abstract

To determine the number of modes in a one dimensional distribution, we compute for each m the Kolmogorov distance between the empirical distribution ${D_m} = {\text{in}}{{\text{f}}_{F \in {u_m}}}{\text{ su}}{{\text{p}}_x}\left| {{F_n}\left( x \right) - F\left( x \right)} \right|$ F _n , and a distribution F ∈ U _m ,the set of uniform mixtures with at most m modes. The m — 1 antimodes obtained in the best fitting m—modal distributions, collected for all m,form a hierarchical tree of intervals. To decide which of these empirical antimodes corresponds to an antimode in the true distribution, we define a test statistic based on a ‘shoulder interval’, a maximal interval of constant density that is neither a mode nor an antimode, in a best fitting m-modal uniform mixture. For each empirical antimode, there is a shoulder interval of minimum length including it, and the points in this shoulder interval are taken as a reference set for evaluating the antimode. The statistic is the maximum deviation of the empirical distribution from a monotone density fit in the shoulder interval, and the reference distribution for it is the maximum empirical excursion for a sample from the uniform with the same sample size as the number of points in the shoulder interval. We demonstrate that this reference distribution gives approximately valid significance tests for a range of population distributions, including some with several modes.