# Unimodal distribution

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)