Difference between revisions of "Unimodal distribution"
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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 <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. | ||
− | 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 [[ | + | 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 |
<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> | <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> |
Revision as of 12:02, 20 October 2012
single-peak distribution
A probability measure on the line whose distribution function is convex for
and concave for
for a certain real
. The number
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 to be the characteristic function of a unimodal distribution with mode at zero it is necessary and sufficient that it admits a representation
![]() |
where is a characteristic function. In terms of distribution functions this equation is equivalent to
![]() |
where and
correspond to
and
. In other words,
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
.
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 is the probability distribution with an atom of size
at
and a density
![]() |
then the density of the convolution of 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 to the point
,
, is called unimodal if there exists an integer
such that
, as a function of
, is non-decreasing for
and non-increasing for
. 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 having a unimodal distribution may be sharpened as follows:
![]() |
for any , where
is the mode and
.
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) |
Unimodal distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodal_distribution&oldid=28560