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''single-peak distribution''
 
''single-peak distribution''
  
A [[Probability measure|probability measure]] on the line whose distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953301.png" /> is convex for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953302.png" /> and concave for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953303.png" /> for a certain real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953304.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953305.png" /> in this case is called the [[Mode|mode]] (peak) and is, generally speaking, not uniquely determined; more precisely, the set of modes of a given unimodal distribution forms a closed interval, possibly degenerate.
+
A [[Probability measure|probability measure]] on the line whose distribution function $  F ( x) $
 +
is convex for $  x < a $
 +
and concave for $  x > a $
 +
for a certain real $  a $.  
 +
The number $  a $
 +
in this case is called the [[Mode|mode]] (peak) and is, generally speaking, not uniquely determined; more precisely, the set of modes of a given unimodal distribution forms a closed interval, possibly degenerate.
  
Examples of unimodal distributions include the [[Normal distribution|normal distribution]], the [[Uniform distribution|uniform distribution]], the [[Cauchy distribution|Cauchy distribution]], the [[Student distribution|Student distribution]], and the [[Chi-squared distribution| "chi-squared"  distribution]]. A.Ya. Khinchin [[#References|[1]]] has obtained the following unimodality criterion: For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953306.png" /> to be the [[Characteristic function|characteristic function]] of a unimodal distribution with mode at zero it is necessary and sufficient that it admits a representation
+
Examples of unimodal distributions include the [[Normal distribution|normal distribution]], the [[Uniform distribution|uniform distribution]], the [[Cauchy distribution|Cauchy distribution]], the [[Student distribution|Student distribution]], and the [[Chi-squared distribution| "chi-squared"  distribution]]. A.Ya. Khinchin [[#References|[1]]] has obtained the following unimodality criterion: For a function $  f $
 +
to be the [[Characteristic function|characteristic function]] of a unimodal distribution with mode at zero it is necessary and sufficient that it admits a representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953307.png" /></td> </tr></table>
+
$$
 +
f ( t)  = {
 +
\frac{1}{t}
 +
} \int\limits _ { 0 } ^ { t }  \phi ( u)  du,\ \
 +
f ( 0= 1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953308.png" /> is a characteristic function. In terms of distribution functions this equation is equivalent to
+
where $  \phi $
 +
is a characteristic function. In terms of distribution functions this equation is equivalent to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u0953309.png" /></td> </tr></table>
+
$$
 +
F ( x)  = \
 +
\int\limits _ { 0 } ^ { 1 }  G \left ( {
 +
\frac{x}{u}
 +
} \right )  du,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533011.png" /> correspond to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533013.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533014.png" /> is unimodal with mode at zero if and only if it is the distribution function of the product of two independent random variables one of which has a [[Uniform distribution|uniform distribution]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533015.png" />.
+
where $  F $
 +
and $  G $
 +
correspond to $  f $
 +
and $  \phi $.  
 +
In other words, $  F $
 +
is unimodal with mode at zero if and only if it is the distribution function of the product of two independent random variables one of which has a [[Uniform distribution|uniform distribution]] on $  [ 0, 1] $.
  
For a distribution given by its characteristic function (as e.g. for a [[Stable distribution|stable distribution]]) the proof of its unimodality presents a difficult analytical problem. The seemingly natural way of representing a given distribution as a limit of unimodal distributions does not achieve this aim, because in general the convolution of two unimodal distributions is not a unimodal distribution (although for symmetric distributions unimodality is preserved under convolution; for a long time it was assumed that this would be so in general). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533016.png" /> is the probability distribution with an atom of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533017.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533018.png" /> and a density
+
For a distribution given by its characteristic function (as e.g. for a [[Stable distribution|stable distribution]]) the proof of its unimodality presents a difficult analytical problem. The seemingly natural way of representing a given distribution as a limit of unimodal distributions does not achieve this aim, because in general the convolution of two unimodal distributions is not a unimodal distribution (although for symmetric distributions unimodality is preserved under convolution; for a long time it was assumed that this would be so in general). For example, if $  F $
 +
is the probability distribution with an atom of size $  1/6 $
 +
at $  5/6 $
 +
and a density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533019.png" /></td> </tr></table>
+
$$
 +
p ( x)  = \left \{
  
then the density of the convolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533020.png" /> with itself has two maxima. Therefore the concept of strong unimodality has been introduced (cf. [[#References|[2]]]); a distribution is said to be strongly unimodal if its convolution with any unimodal distribution is unimodal. Every strongly unimodal distribution is unimodal.
+
then the density of the convolution of $  F $
 +
with itself has two maxima. Therefore the concept of strong unimodality has been introduced (cf. [[#References|[2]]]); a distribution is said to be strongly unimodal if its convolution with any unimodal distribution is unimodal. Every strongly unimodal distribution is unimodal.
  
A [[Lattice distribution|lattice distribution]] giving probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533021.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533023.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533024.png" />, is called unimodal if there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533026.png" />, as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533027.png" />, is non-decreasing for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533028.png" /> and non-increasing for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533029.png" />. Examples of unimodal lattice distributions are the [[Poisson distribution|Poisson distribution]], the [[Binomial distribution|binomial distribution]] and the [[Geometric distribution|geometric distribution]].
+
A [[Lattice distribution|lattice distribution]] giving probability $  p _ {k} $
 +
to the point $  a + hk $,
 +
$  k = 0, \pm  1 , \pm  2 \dots $
 +
$  h > 0 $,  
 +
is called unimodal if there exists an integer $  k _ {0} $
 +
such that $  p _ {k} $,  
 +
as a function of $  k $,  
 +
is non-decreasing for $  k \leq  k _ {0} $
 +
and non-increasing for $  k \geq  k _ {0} $.  
 +
Examples of unimodal lattice distributions are the [[Poisson distribution|Poisson distribution]], the [[Binomial distribution|binomial distribution]] and the [[Geometric distribution|geometric distribution]].
  
Certain results concerning distributions may be strengthened by assuming unimodality. E.g. the [[Chebyshev inequality in probability theory|Chebyshev inequality in probability theory]] for a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533030.png" /> having a unimodal distribution may be sharpened as follows:
+
Certain results concerning distributions may be strengthened by assuming unimodality. E.g. the [[Chebyshev inequality in probability theory|Chebyshev inequality in probability theory]] for a random variable $  \xi $
 +
having a unimodal distribution may be sharpened as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533031.png" /></td> </tr></table>
+
$$
 +
{\mathsf P}
 +
\{ | \xi - x _ {0} | \geq  k \zeta \}  \leq  {
 +
\frac{4}{9k  ^ {2} }
 +
}
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533033.png" /> is the mode and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095330/u09533034.png" />.
+
for any $  k > 0 $,  
 +
where $  x _ {0} $
 +
is the mode and $  \zeta  ^ {2} = {\mathsf E} ( \xi - x _ {0} )  ^ {2} $.
  
 
====References====
 
====References====

Revision as of 08:27, 6 June 2020


single-peak distribution

A probability measure on the line whose distribution function $ F ( x) $ is convex for $ x < a $ and concave for $ x > a $ for a certain real $ a $. The number $ a $ in this case is called the mode (peak) and is, generally speaking, not uniquely determined; more precisely, the set of modes of a given unimodal distribution forms a closed interval, possibly degenerate.

Examples of unimodal distributions include the normal distribution, the uniform distribution, the Cauchy distribution, the Student distribution, and the "chi-squared" distribution. A.Ya. Khinchin [1] has obtained the following unimodality criterion: For a function $ f $ to be the characteristic function of a unimodal distribution with mode at zero it is necessary and sufficient that it admits a representation

$$ f ( t) = { \frac{1}{t} } \int\limits _ { 0 } ^ { t } \phi ( u) du,\ \ f ( 0) = 1, $$

where $ \phi $ is a characteristic function. In terms of distribution functions this equation is equivalent to

$$ F ( x) = \ \int\limits _ { 0 } ^ { 1 } G \left ( { \frac{x}{u} } \right ) du, $$

where $ F $ and $ G $ correspond to $ f $ and $ \phi $. In other words, $ F $ is unimodal with mode at zero if and only if it is the distribution function of the product of two independent random variables one of which has a uniform distribution on $ [ 0, 1] $.

For a distribution given by its characteristic function (as e.g. for a stable distribution) the proof of its unimodality presents a difficult analytical problem. The seemingly natural way of representing a given distribution as a limit of unimodal distributions does not achieve this aim, because in general the convolution of two unimodal distributions is not a unimodal distribution (although for symmetric distributions unimodality is preserved under convolution; for a long time it was assumed that this would be so in general). For example, if $ F $ is the probability distribution with an atom of size $ 1/6 $ at $ 5/6 $ and a density

$$ p ( x) = \left \{ then the density of the convolution of $ F $ with itself has two maxima. Therefore the concept of strong unimodality has been introduced (cf. [[#References|[2]]]); a distribution is said to be strongly unimodal if its convolution with any unimodal distribution is unimodal. Every strongly unimodal distribution is unimodal. A [[Lattice distribution|lattice distribution]] giving probability $ p _ {k} $ to the point $ a + hk $, $ k = 0, \pm 1 , \pm 2 \dots $ $ h > 0 $, is called unimodal if there exists an integer $ k _ {0} $ such that $ p _ {k} $, as a function of $ k $, is non-decreasing for $ k \leq k _ {0} $ and non-increasing for $ k \geq k _ {0} $. Examples of unimodal lattice distributions are the [[Poisson distribution|Poisson distribution]], the [[Binomial distribution|binomial distribution]] and the [[Geometric distribution|geometric distribution]]. Certain results concerning distributions may be strengthened by assuming unimodality. E.g. the [[Chebyshev inequality in probability theory|Chebyshev inequality in probability theory]] for a random variable $ \xi $ having a unimodal distribution may be sharpened as follows: $$ {\mathsf P} \{ | \xi - x _ {0} | \geq k \zeta \} \leq { \frac{4}{9k ^ {2} }

}

$$

for any $ k > 0 $, where $ x _ {0} $ is the mode and $ \zeta ^ {2} = {\mathsf E} ( \xi - x _ {0} ) ^ {2} $.

References

[1] A.Ya. Khinchin, "On unimodal distributions" Izv. Nauk Mat. i Mekh. Inst. Tomsk, 2 : 2 (1938) pp. 1–7 (In Russian)
[2] I.A. Ibragimov, "On the composition of unimodal distributions" Theor. Probab. Appl., 1 : 2 (1956) pp. 255–260 Teor. Veroyatnost. i Primenen., 1 : 2 (1956) pp. 283–288
[3] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)

Comments

A non-degenerate strongly unimodal distribution has a log-concave density.

References

[a1] S. Dharmadhikari, K. Yong-Dev, "Unimodality, convexity, and applications" , Acad. Press (1988)
How to Cite This Entry:
Unimodal distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodal_distribution&oldid=49076
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article