Davis distribution

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Davis distribution
Probability density function No image available
Cumulative distribution function No image available
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

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, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
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
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' 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 SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete 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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