Editing Multimodal distribution (section)

===Unimodal vs. bimodal distribution===

Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions.<ref name=Pearson1894>{{cite journal | last1 = Pearson | first1 = K | year = 1894 | title = Contributions to the mathematical theory of evolution: On the dissection of asymmetrical frequency-curves | journal = Philosophical Transactions of the Royal Society A | volume = 185 | pages = 71–90 | doi=10.1098/rsta.1894.0003| bibcode = 1894RSPTA.185...71P| doi-access = free }}</ref> This method required the solution of a ninth order [[polynomial]]. In a subsequent paper Pearson reported that for any distribution skewness<sup>2</sup> + 1 < kurtosis.<ref name=Pearson1916/> Later Pearson showed that<ref name=Pearson1929>{{cite journal | last1 = Pearson | first1 = K | year = 1929 | title = Editorial note | journal = Biometrika | volume = 21 | pages = 370–375 }}</ref>

<math display="block"> b_2 - b_1 \ge 1 </math>

where ''b''<sub>2</sub> is the kurtosis and ''b''<sub>1</sub> is the square of the skewness. Equality holds only for the two point [[Bernoulli distribution]] or the sum of two different [[Dirac delta function]]s. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1.

Baker proposed a transformation to convert a bimodal to a unimodal distribution.<ref name=Baker1930>{{cite journal | last1 = Baker | first1 = GA | year = 1930 | title = Transformations of bimodal distributions | journal = Annals of Mathematical Statistics | volume = 1 | issue = 4| pages = 334–344 | doi=10.1214/aoms/1177733063| doi-access = free }}</ref>

Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences.<ref name=Haldane1951>{{cite journal | last1 = Haldane | first1 = JBS | year = 1951 | title = Simple tests for bimodality and bitangentiality | journal = Annals of Eugenics  | volume = 16 | issue = 1| pages = 359–364 | doi = 10.1111/j.1469-1809.1951.tb02488.x | pmid = 14953132 }}</ref> Larkin later introduced a test based on the F test;<ref name=Larkin1979>{{cite journal | last1 = Larkin | first1 = RP | year = 1979 | title = An algorithm for assessing bimodality vs. unimodality in a univariate distribution | journal = Behavior Research Methods & Instrumentation | volume = 11 | issue = 4| pages = 467–468 | doi = 10.3758/BF03205709 | doi-access = free }}</ref> Benett created one based on [[G-test|Fisher's G test]].<ref name=Bennett1992>{{cite journal | last1 = Bennett | first1 = SC | year = 1992 | title = Sexual dimorphism of ''Pteranodon'' and other pterosaurs, with comments on cranial crests | journal = Journal of Vertebrate Paleontology | volume = 12 | issue = 4| pages = 422–434 | doi=10.1080/02724634.1992.10011472}}</ref> Tokeshi has proposed a fourth test.<ref name=Tokeshi1992>{{cite journal | last1 = Tokeshi | first1 = M | year = 1992 | title = Dynamics and distribution in animal communities; theory and analysis | journal = Researches on Population Ecology | volume = 34 | issue = 2| pages = 249–273 | doi=10.1007/bf02514796| s2cid = 22912914 }}</ref><ref name=Barreto2003>{{cite journal | last1 = Barreto | first1 = S | last2 = Borges | first2 = PAV | last3 = Guo | first3 = Q | year = 2003 | title = A typing error in Tokeshi's test of bimodality | journal = Global Ecology and Biogeography | volume = 12 | issue = 2| pages = 173–174 | doi=10.1046/j.1466-822x.2003.00018.x| hdl = 10400.3/1408 | hdl-access = free }}</ref> A test based on a likelihood ratio has been proposed by Holzmann and Vollmer.<ref name=Holzmann2008/>

A method based on the score and Wald tests has been proposed.<ref name=Carolan2001>{{cite journal | last1 = Carolan | first1 = AM | last2 = Rayner | first2 = JCW | year = 2001 | title = One sample tests for the location of modes of nonnormal data | journal =  Journal of Applied Mathematics and Decision Sciences| volume = 5 | issue = 1| pages = 1–19 | doi=10.1155/s1173912601000013| citeseerx = 10.1.1.504.4999 | doi-access = free }}</ref> This method can distinguish between unimodal and bimodal distributions when the underlying distributions are known.