The Dagum distribution (or Mielke Beta-Kappa distribution) is a continuous
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
defined over
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
. It is named after Camilo Dagum, who proposed it in a series of papers in the 1970s.
The Dagum distribution arose from several variants of a new model on the size distribution of personal income and is mostly associated with the study of
income distribution
In economics, income distribution covers how a country's total GDP is distributed amongst its population. Economic theory and economic policy have long seen income and its distribution as a central concern. Unequal distribution of income causes eco ...
. There is both a three-parameter specification (Type I) and a four-parameter specification (Type II) of the Dagum distribution; a summary of the genesis of this distribution can be found in "A Guide to the Dagum Distributions".
A general source on statistical size distributions often cited in work using the Dagum distribution is ''Statistical Size Distributions in Economics and Actuarial Sciences''.
Definition
The
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of the Dagum distribution (Type I) is given by
:
The corresponding
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
is given by
:
The
quantile function
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equ ...
is given by
:
The Dagum distribution can be derived as a special case of the
generalized Beta II (GB2) distribution (a generalization of the
Beta prime distribution
In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution.
Definitions
...
):
:
There is also an intimate relationship between the Dagum and
Singh–Maddala / Burr distribution.
:
The
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of the Dagum (Type II) distribution adds a point mass at the origin and then follows a Dagum (Type I) distribution over the rest of the support (i.e. over the positive halfline)
:
Use in economics
The Dagum distribution is often used to model income and wealth distribution. The relation between the Dagum Type I and the
gini coefficient is summarized in the formula below:
:
where
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. Note that this value is independent from the scale-parameter,
.
Although the Dagum distribution is not the only three parameter distribution used to model income distribution it is usually the most appropriate.
References
External links
Camilo Dagum (1925 - 2005): obituary
{{ProbDistributions, continuous-semi-infinite
Continuous distributions
Income inequality metrics
Economic inequality