Bates distribution

Bates

Parameters	$-\infty <a<b<\infty \,$ $n \geq 1$ integer
Support	$x\in [a,b]$
PDF	see below
Mean	${\tfrac {1}{2}}(a+b)$
Variance	${\tfrac {1}{12n}}(b-a)^{2}$
Skewness	0
Ex. kurtosis	$-\tfrac{6}{5n}$
CF	$\left(-{\frac {in(e^{{{\tfrac {ibt}{n}}}}-e^{{{\tfrac {iat}{n}}}})}{(b-a)t}}\right)^{n}$

In probability and statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval.^[1] This distribution is sometimes confused with the Irwin–Hall distribution, which is the distribution of the sum (not mean) of n independent random variables uniformly distributed from 0 to 1.

Definition

The Bates distribution is the continuous probability distribution of the mean, X, of n independent uniformly distributed random variables on the unit interval, U_i:

$X={\frac {1}{n}}\sum _{{k=1}}^{n}U_{k}.$

The equation defining the probability density function of a Bates distribution random variable x is

$f_{X}(x;n)={\frac {n}{2\left(n-1\right)!}}\sum _{{k=0}}^{{n}}\left(-1\right)^{k}{n \choose k}\left(nx-k\right)^{{n-1}}\operatorname{sgn}(nx-k)$

for x in the interval (0,1), and zero elsewhere. Here sgn(x − k) denotes the sign function:

$\operatorname{sgn} \left(nx-k\right)={\begin{cases}-1&nx<k\\0&nx=k\\1&nx>k.\end{cases}}$

More generally, the mean of n independent uniformly distributed random variables on the interval [a,b]

$X_{{(a,b)}}={\frac {1}{n}}\sum _{{k=1}}^{n}U_{k}(a,b).$

would have the probability density function (PDF) of

$g(x;n,a,b)=f_{X}\left({\frac {x-a}{b-a}};n\right){\text{ for }}a\leq x\leq b\,$

Therefore, the PDF of the distribution is $PDF = \begin{cases} \sum_{k=0}^n (-1)^k \binom{n}{k} \left( \frac{x-a}{b-a}-k/n \right)^{n-1} \sgn\left( \frac{x-a}{b-a}-k/n \right) & \text{if } x\in[a,b]\\ 0 & \text{otherwise} \end{cases}$

Extensions to the Bates Distribution

Instead of dividing by n we can also use sqrt(n) and create this way a similar distribution with a constant variance (like unity) can be created. With subtraction of the mean we can set the resulting mean of zero. This way the parameter n would become a purely shape adjusting parameter, and we obtain a distribution which covers the uniform, the triangular and in the limit also the normal Gaussian distribution. By allowing also non-integer n a highly flexible distribution can be created (e.g. U(0,1)+0.5U(0,1) gives a trapezodial distribution). Actually the Student-t distribution provides a natural extension of the normal Gaussian distribution for modeling of long tail data. And such generalized Bates distribution is doing so for short tail data (kurtosis<3).

Notes

1. Jonhson, N.L.; Kotz, S.; Balakrishnan (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)

References

• Bates,G.E. (1955) "Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya urn scheme", Annals of Mathematical Statistics, 26, 705–720