Editing Unimodality (section)

===Inequalities===
{{See also|Chebyshev's inequality#Unimodal distributions}}

====Gauss's inequality====

A first important result is [[Gauss's inequality]].<ref>{{cite journal|last=Gauss|first=C. F.|author-link=Carl Friedrich Gauss|year=1823|title=Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior|journal=Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores|volume=5}}</ref> Gauss's inequality gives an upper bound on the probability that a value lies more than any given distance from its mode. This inequality depends on unimodality.

====Vysochanskiï–Petunin inequality====

A second is the [[Vysochanskiï–Petunin inequality]],<ref>{{cite journal |author=D. F. Vysochanskij, Y. I. Petunin |year=1980 |title=Justification of the 3σ rule for unimodal distributions |journal=Theory of Probability and Mathematical Statistics |volume=21 |pages=25–36}}</ref> a refinement of the [[Chebyshev inequality]]. The Chebyshev inequality guarantees that in any probability distribution, "nearly all" the values are "close to" the mean value. The Vysochanskiï–Petunin inequality refines this to even nearer values, provided that the distribution function is continuous and unimodal. Further results were shown by Sellke and Sellke.<ref>{{Cite journal
 | last1 = Sellke | first1 =  T.M.
 | last2 =  Sellke	 | first2 =  S.H.
 | title = Chebyshev inequalities for unimodal distributions
 | jstor = 2684690
 | year = 1997
 | journal = [[American Statistician]]
 | volume = 51
 | issue = 1
 | pages = 34–40
 | publisher = American Statistical Association
 | doi=10.2307/2684690
}}</ref>

====Mode, median and mean====

Gauss also showed in 1823 that for a unimodal distribution<ref name=Gauss1823>Gauss C.F. Theoria Combinationis Observationum Erroribus Minimis Obnoxiae. Pars Prior. Pars Posterior. Supplementum. Theory of the Combination of Observations Least Subject to Errors. Part One. Part Two. Supplement. 1995. Translated by G.W. Stewart. Classics in Applied Mathematics Series, Society for Industrial and Applied Mathematics, Philadelphia</ref>

: <math>\sigma \le \omega \le 2 \sigma</math>

and

: <math>|\nu - \mu| \le \sqrt{\frac{3}{4}} \omega ,</math>

where the [[median]] is ''ν'', the mean is ''μ'' and  ''ω'' is the [[root mean square deviation]] from the mode.

It can be shown for a unimodal distribution that the median ''ν'' and the  mean ''μ'' lie within (3/5)<sup>1/2</sup> ≈ 0.7746 [[standard deviation]]s of each other.<ref name="unimodal">{{cite journal | url=http://epubs.siam.org/doi/pdf/10.1137/S0040585X97975447 | doi=10.1137/S0040585X97975447 | title=The Mean, Median, and Mode of Unimodal Distributions: A Characterization | year=1997 | last1=Basu | first1=S. | last2=Dasgupta | first2=A. | journal=Theory of Probability & Its Applications | volume=41 | issue=2 | pages=210–223 }}</ref> In symbols,

: <math>\frac{|\nu - \mu|}{\sigma} \le \sqrt{\frac{3}{5}}</math>

where | . | is the [[absolute value]].

In 2020, Bernard, Kazzi, and Vanduffel generalized the previous inequality by deriving the maximum distance between the symmetric quantile average <math>\frac{ q_\alpha  + q_{(1-\alpha)} }{ 2 } </math> and the mean,<ref name="unimodalbounds">{{cite journal | doi=10.1016/j.insmatheco.2020.05.013 | title=Range Value-at-Risk bounds for unimodal distributions under partial information | year=2020 | last1=Bernard | first1=Carole | last2=Kazzi | first2=Rodrigue | last3=Vanduffel | first3=Steven | journal=Insurance: Mathematics and Economics | volume=94 | pages=9–24 | doi-access=free }}</ref>

: <math>\frac{ \left| \frac{ q_\alpha  + q_{(1-\alpha)} }{2}  - \mu \right| }{ \sigma } \le \left\{
	\begin{array}{cl}
	\frac{\sqrt[]{\frac{4}{9(1-\alpha)}-1} \text{ } + \text{ } \sqrt[]{\frac{1-\alpha}{1/3+\alpha}}}{2}
	& \text{for }\alpha \in  \left[\frac{5}{6},1\right)\!, \\ 
	\frac{\sqrt[]{\frac{3 \alpha}{4-3\alpha}} \text{ } + \text{ } \sqrt[]{\frac{1-\alpha}{1/3+\alpha}}}{2}
	& \text{for }\alpha \in \left(\frac{1}{6},\frac{5}{6}\right)\!,\\
	\frac{\sqrt[]{\frac{3 \alpha}{4-3\alpha}} \text{ } + \text{ } \sqrt[]{\frac{4}{9 \alpha} -1}}{2}
	& \text{for }\alpha \in  \left(0,\frac{1}{6}\right]\!.
	\end{array}
	\right.</math>

The maximum distance is minimized at <math>\alpha=0.5</math> (i.e., when the symmetric quantile average is equal to  <math>q_{0.5} = \nu</math>), which indeed motivates the common choice of the median as a robust estimator for the mean. Moreover, when <math>\alpha = 0.5</math>, the bound is equal to <math>\sqrt{3/5}</math>, which is the maximum distance between the median and the mean of a unimodal distribution.

A similar relation holds between the median and the mode ''θ'': they lie within 3<sup>1/2</sup> ≈ 1.732 standard deviations of each other:

: <math>\frac{|\nu - \theta|}{\sigma} \le \sqrt{3}.</math>

It can also be shown that the mean and the mode lie within 3<sup>1/2</sup> of each other:

: <math>\frac{|\mu - \theta|}{\sigma} \le \sqrt{3}.</math>

====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}.