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)

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)