Abstract
In the univariate setting a distribution function (d.f.) F is unimodal with mode m if F is convex in (-∞, m] and concave in (m, ∞). Hartigan & Hartigan (1985) proposed the DIP statistic for testing whether a distribution is unimodal against a general multimodal alternative. The dip of a d.f. F is defined to be the maximum difference between F and the unimodal distribution function that minimises that maximum difference. i.e. the dip of a distribution function F is: $$ DIP(F) = \mathop {\inf }\limits_{G \in \Lambda } \mathop {\sup }\limits_x |F(x) - G(x)| $$ where A is the class of all distributions with unimodal density functions.
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