Editing Multimodal distribution (section)

===General tests===

To test if a distribution is other than unimodal, several additional tests have been devised: the [[bandwidth test (multimodal)|bandwidth test]],<ref name=Silverman1981/> the [[dip test]],<ref name=Hartigan1985>{{cite journal | last1 = Hartigan | first1 = JA | last2 = Hartigan | first2 = PM | year = 1985 | title = The dip test of unimodality | journal = Annals of Statistics | volume = 13 | issue = 1| pages = 70–84 | doi=10.1214/aos/1176346577| doi-access = free }}</ref> the [[excess mass test]],<ref name=Mueller1991>{{cite journal | last1 = Mueller | first1 = DW | last2 = Sawitzki | first2 = G | year = 1991 | title = Excess mass estimates and tests for multimodality | journal = Journal of the American Statistical Association | volume = 86 | issue = 415| pages = 738–746 |jstor=2290406 | doi=10.1080/01621459.1991.10475103}}</ref> the MAP test,<ref name="Rozál1994">{{cite journal | last1 = Rozál | first1 = GPM Hartigan JA | year = 1994 | title = The MAP test for multimodality | journal = Journal of Classification | volume = 11 | issue = 1| pages = 5–36 | doi = 10.1007/BF01201021 | s2cid = 118500771 }}</ref> the [[mode existence test]],<ref name=Minnotte1997>{{cite journal | last1 = Minnotte | first1 = MC | year = 1997 | title = Nonparametric testing of the existence of modes | journal = Annals of Statistics | volume = 25 | issue = 4| pages = 1646–1660 | doi=10.1214/aos/1031594735| doi-access = free }}</ref> the [[runt test]],<ref name=Hartigan1992>{{cite journal | last1 = Hartigan | first1 = JA | last2 = Mohanty | first2 = S | year = 1992 | title = The RUNT test for multimodality | journal = Journal of Classification | volume = 9 | pages = 63–70 | doi=10.1007/bf02618468| s2cid = 121960832 }}</ref><ref name=Andrushkiw2008>{{cite journal |author1=Andrushkiw RI |author2=Klyushin DD |author3=Petunin YI |date=2008 |title=A new test for unimodality |journal=Theory of Stochastic Processes |volume=14 |issue=1 |pages=1–6}}</ref> the [[span test]],<ref name=Hartigan1988>{{cite book |last=Hartigan |first=J. A. |year=1988 |chapter=The Span Test of Multimodality |title=Classification and Related Methods of Data Analysis |editor-first=H. H. |editor-last=Bock |publisher=North-Holland |location=Amsterdam |pages=229–236 |isbn=0-444-70404-3 }}</ref> and the [[saddle test]].

An implementation of the dip test is available for the [[R (programming language)|R programming language]].<ref>{{cite web|url=https://cran.r-project.org/web/packages/diptest/index.html|title=diptest: Hartigan's Dip Test Statistic for Unimodality - Corrected|first1=Martin Maechler (originally from Fortran and S.-plus by Dario|last1=Ringach|last2=NYU.edu)|date=5 December 2016|via=R-Packages}}</ref> The p-values for the dip statistic values range between 0 and 1. P-values less than 0.05 indicate significant multimodality and p-values greater than 0.05 but less than 0.10 suggest multimodality with marginal significance.<ref name=FreemanDale2012>{{cite journal | last1 = Freeman | last2 = Dale | year = 2012 | title = Assessing bimodality to detect the presence of a dual cognitive process | journal = Behavior Research Methods | volume = 45 | issue = 1 | pages = 83–97 | doi = 10.3758/s13428-012-0225-x | pmid = 22806703 | s2cid = 14500508 | url = http://psych.nyu.edu/freemanlab/pubs/2012_BRM.pdf| doi-access = free }}</ref>