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
Excess 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. 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 $PDF={\begin{cases}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}\left({\frac {x-a}{b-a}}-k/n\right)^{n-1}\operatorname {sgn} \left({\frac {x-a}{b-a}}-k/n\right)&{\text{if }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}$

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 substraction 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).

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