statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the Davis distributions are a family of

continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

s. 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 In physics, Planck's law (also Planck radiation law) describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the ...

of radiation from

statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

Definition

The

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

of the Davis distribution is given by :

f(x;\mu,b,n)=\frac

where

\Gamma(n)

is the

Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

and

\zeta(n)

is the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

. 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 Kleiber 2003 *

f(\mu)=0 \,

for some

\mu>0 \,

* A modal income exists * For large ''x'', the density behaves like a

Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial scien ...

: ::

f(x) \sim A ^ \, .

Related distributions

* If

X \sim \mathrm(b=1,n=4,\mu=0)\,

then

\tfrac \sim \mathrm

(

)

Notes

References

* *Davis, H. T. (1941)
''The Analysis of Economic Time Series''
The Principia Press, Bloomington, Indian
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) {{ProbDistributions, continuous-semi-infinite Continuous distributions