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}\sgn(nx-k)

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

\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 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).

Notes

↑ 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

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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