Editing Unimodality (section)

====Skewness and kurtosis====

Rohatgi and Szekely claimed that the [[skewness]] and [[kurtosis]] of a unimodal distribution are related by the inequality:<ref name=Rohatgi1989>{{cite journal | doi=10.1016/0167-7152(89)90035-7 | title=Sharp inequalities between skewness and kurtosis | year=1989 | last1=Rohatgi | first1=Vijay K. | last2=Székely | first2=Gábor J. | journal=Statistics & Probability Letters | volume=8 | issue=4 | pages=297–299 }}</ref>

: <math> \gamma^2 - \kappa \le \frac{ 6 }{ 5 } = 1.2 </math>

where ''κ'' is the kurtosis and ''γ'' is the skewness. Klaassen, Mokveld, and van Es showed that this only applies in certain settings, such as the set of unimodal distributions where the mode and mean coincide.<ref name=Klaassen2000>{{cite journal | doi=10.1016/S0167-7152(00)00090-0 | title=Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions | year=2000 | last1=Klaassen | first1=Chris A.J. | last2=Mokveld | first2=Philip J. | last3=Van Es | first3=Bert | journal=Statistics & Probability Letters | volume=50 | issue=2 | pages=131–135 }}</ref>

They derived a weaker inequality which applies to all unimodal distributions:<ref name=Klaassen2000 />

: <math> \gamma^2 - \kappa \le \frac{ 186 }{ 125 } = 1.488 </math>

This bound is sharp, as it is reached by the equal-weights mixture of the uniform distribution on [0,1] and the discrete distribution at {0}.