Bates distribution

Definition

The Bates distribution on the unit interval [0,1] and with parameter n \in \mathbb N is the continuous probability distribution of the empirical mean X of n independent random variables U_1,...,U_n uniformly distributed on the unit interval:

: X = \frac{1}{n}\sum_{k=1}^n U_k.

The probability density function is

: f_X(x; n)= \frac n {2(n-1)!} \sum_{k=0}^n (-1)^k {n \choose k} (nx-k)^{n-1} \sgn(nx-k)

for x in the open interval ]0,1 , and zero elsewhere. Here \sgn(nx-k) denotes the [sign function:

: \sgn(nx-k) = \begin{cases} -1 & nx 0 & nx = k \ 1 & nx k. \end{cases}

More generally, the empirical mean X^{(a,b)} of n independent random variables U_1^{(a,b)},...,U_n^{(a,b)} uniformly distributed on the interval [a,b]

: X^{(a,b)} = \frac{1}{n}\sum_{k=1}^n U_k^{(a,b)}

has the following the probability density function (PDF) of

: f_{X^{(a,b)}} (x; n) = \frac{1}{b-a} f_X\left(\frac{x-a}{b-a};n\right) for x \in ]a,b and zero otherwise.

Extensions and Applications

With a few modifications, the Bates distribution encompasses the [uniform, the triangular, and, taking the limit as n goes to infinity, also the normal Gaussian distribution.

Replacing the term \frac{1}{n} when calculating the mean, X, with \frac{1}{\sqrt{n}} will create a similar distribution with a constant variance, such as unity. Then, by subtracting the mean, the resulting mean of the distribution will be set at zero. Thus the parameter n would become a purely shape-adjusting parameter. By also allowing n to be a non-integer, a highly flexible distribution can be created, for example, U(0,1) + 0.5U(0,1) gives a trapezoidal distribution.

The Student-t distribution provides a natural extension of the normal Gaussian distribution for modeling of long tail data. A Bates distribution that has been generalized as previously stated fulfills the same purpose for short tail data.

The Bates distribution has an application to beamforming and pattern synthesis in the field of electrical engineering. The distribution was found to increase the beamwidth of the main lobe, representing an increase in the signal of the radiation pattern in a single direction, while simultaneously reducing the usually undesirable sidelobe levels.

References

References

1. (1995). "Continuous Univariate Distributions, Volume 2". John Wiley & Sons.
2. "The thing named "Irwin-Hall distribution" in d3.random is actually a Bates distribution · Issue #1647 · d3/d3".
3. (January 2018). "Sidelobe behavior and bandwidth characteristics of distributed antenna arrays".
4. (2003). "Encyclopedia of Physical Science and Technology".