Davis distribution

Davis distribution
Parameters	$b>0$ scale $n>0$ shape $\mu >0$ location
Support	$x>\mu$
PDF	${\frac {b^{n}{(x-\mu )}^{-1-n}}{\left(e^{\frac {b}{x-\mu }}-1\right)\Gamma (n)\zeta (n)}}$ Where $\Gamma (n)$ is the Gamma function and $\zeta (n)$ is the Riemann zeta function
Mean	${\begin{cases}\mu +{\frac {b\zeta (n-1)}{(n-1)\zeta (n)}}&{\text{if}}\ n>2\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}$
Variance	${\begin{cases}{\frac {b^{2}\left(-(n-2){\zeta (n-1)}^{2}+(n-1)\zeta (n-2)\zeta (n)\right)}{(n-2){(n-1)}^{2}{\zeta (n)}^{2}}}&{\text{if}}\ n>3\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}$

In statistics, the Davis distributions are a family of continuous probability distributions. It is named after Harold T. Davis (1892–1974), who in 1941 proposed this distribution to model income sizes. (The Theory of Econometrics and Analysis of Economic Time Series). It is a generalization of the Planck's law of radiation from statistical physics.

Definition

The probability density function of the Davis distribution is given by

f(x;\mu ,b,n)={\frac {b^{n}{(x-\mu )}^{-1-n}}{\left(e^{\frac {b}{x-\mu }}-1\right)\Gamma (n)\zeta (n)}}

where $\Gamma (n)$ is the Gamma function and $\zeta (n)$ is the Riemann zeta function. Here μ, b, and n are parameters of the distribution, and n need not be an integer.

Background

In an attempt to derive an expression that would represent not merely the upper tail of the distribution of income, Davis required an appropriate model with the following properties^[1]

$f(\mu )=0\,$ for some $\mu >0\,$
A modal income exists
For large x, the density behaves like a Pareto distribution:

f(x)\sim A{(x-\mu )}^{-\alpha -1}\,.

Related distributions

If $X\sim \mathrm {Davis} (b=1,n=4,\mu =0)\,$ then
${\tfrac {1}{X}}\sim \mathrm {Planck}$ (Planck's law)

Notes

↑ Kleiber 2003

References

Kleiber, Christian (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley Series in Probability and Statistics. ISBN 978-0-471-15064-0.
Davis, H. T. (1941). The Analysis of Economic Time Series. The Principia Press, Bloomington, Indiana Download book
VICTORIA-FESER, Maria-Pia. (1993) Robust methods for personal income distribution models. Thèse de doctorat : Univ. Genève, 1993, no. SES 384 (p. 178)

Probability distributions

List

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 Lomax 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 Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's 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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