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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>
+
$$
 +
p ( x)  = \left \{ ??? \right \}
 +
$$
  
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]] 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]], the [[binomial distribution]] and the [[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]] 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} $
====References====
+
is the mode and $  \zeta  ^ {2} = {\mathsf E} ( \xi - x _ {0} ) ^ {2} $.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.Ya. Khinchin, "On unimodal distributions" ''Izv. Nauk Mat. i Mekh. Inst. Tomsk'', '''2''' : 2 (1938) pp. 1–7 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR></table>
 
  
 
====Comments====
 
====Comments====
 
A non-degenerate strongly unimodal distribution has a log-concave density.
 
A non-degenerate strongly unimodal distribution has a log-concave density.
 +
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Dharmadhikari,  K. Yong-Dev,  "Unimodality, convexity, and applications" , Acad. Press  (1988)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.Ya. Khinchin, "On unimodal distributions" ''Izv. Nauk Mat. i Mekh. Inst. Tomsk'', '''2''' : 2 (1938) pp. 1–7 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Dharmadhikari,  K. Yong-Dev,  "Unimodality, convexity, and applications" , Acad. Press  (1988)</TD></TR></table>

Latest revision as of 07:36, 10 April 2023


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

$$ p ( x) = \left \{ ??? \right \} $$

then the density of the convolution of $ F $ with itself has two maxima. Therefore the concept of strong unimodality has been introduced (cf. [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 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, the binomial distribution and the geometric distribution.

Certain results concerning distributions may be strengthened by assuming unimodality. E.g. the 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} $.

Comments

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


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)
[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=25964
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article