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Dagum 4P distribution

Why it exists

Dagum 4P adds a location parameter to the Dagum distribution. It is useful when a variable starts above a threshold but still needs a flexible economic or reliability tail.

Alternative names

It may also be called four-parameter Dagum or shifted Dagum. In economic comparisons it may appear inside GB2 with an added location.

Constructions and limits

Fixing loc at zero recovers the three-parameter Dagum. It retains links with Burr III, GB2, Beta prime, and loglogistic, but parameter comparisons require the same origin.

Applications

It can separate an entry minimum, such as an age, detectable size, or preparation time, from subsequent variability. In economics, loc should have an institutional interpretation rather than merely forcing a better fit.

Before interpreting inequality or risk, check moment existence after shifting and report which part of the scale belongs to the threshold.

Income or size above an institutional origin

Dagum 4P adds location to the three-parameter economic model. The floor may represent a deductible, an exempt amount, or a minimum recorded size. Beyond that point, the shapes governing inequality and tail behaviour remain.

For income data, a positive or negative loc need not have an economically sensible meaning even when likelihood improves. It may be compensating for debt, omitted zeros, or a population definition incompatible with the model. Location also changes the Lorenz curve and relative inequality measures because those measures are not invariant to adding a constant. The origin should therefore receive an institutional justification, and inequality summaries should be recomputed with uncertainty.

Decision guide

A good candidate when: the Dagum shape is appropriate and the process has an interpretable floor above zero.

Compare it with: unshifted Dagum and 4P Burr. Verify that the estimated floor is not determined almost entirely by one or two minimum observations.

References

  • SciPy reference: scipy.stats.mielke — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Dagum, C. (1977). A new model of personal income distribution: Specification and estimation. Swiss Journal of Economics and Statistics, 113, 185–211.
  • Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley.

Dagum 4P Distribution: equations and calculator

Distribution defintion

X∼Dagum4P(a,b,p,Loc)X\sim\mathrm{Dagum_{4P}}\left(a,b,p,\text{Loc}\right)

Distribution domain

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Parameters domain and parameters constraints

a∈R+,b∈R+,p∈R+,Loc∈Ra\in\mathbb{R}^{+},b\in\mathbb{R}^{+},p\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Cumulative distribution function

FX(x)=(1+(x−Locb)−a)−pF_{X}\left(x\right)={\left(1+{\left(\frac{x-\text{Loc}}{b}\right)}^{-a}\right)}^{-p}

Probability density function

fX(x)=apx−Loc((x−Locb)ap((x−Locb)a+1)p+1)f_{X}\left(x\right)=\frac{a p}{x-\text{Loc}}\left(\frac{(\tfrac{x-\text{Loc}}{b})^{a p}}{\left((\tfrac{x-\text{Loc}}{b})^a+1\right)^{p+1}}\right)

Percent point function/Sample

FX−1(u)=Loc+b(u−1/p−1)−1/aF^{-1}_{X}\left(u\right)=\text{Loc}+b(u^{-1/p}-1)^{-1/a}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=pbk⋅Beta(ap+ka,a−ka)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=pb^{k}\cdot \text{Beta}\left(\frac{ap+k}{a},\frac{a-k}{a}\right)

Parametric mean

Mean(X)=Loc+μ~1′\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}

Parametric variance

Variance(X)=μ~2′−μ~1′2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}

Parametric median

Median(X)=Loc+b(−1+21p)−1a\mathrm{Median}(X)=\text{Loc}+b{\left(-1+2^{\tfrac{1}{p}}\right)}^{-\tfrac{1}{a}}

Parametric mode

Mode(X)=Loc+b(ap−1a+1)1a\mathrm{Mode}(X)=\text{Loc}+b{\left(\frac{ap-1}{a+1}\right)}^{\tfrac{1}{a}}

Additional information and definitions

Xˉ∼Dagum(a,b,p)\bar{X}\sim\mathrm{Dagum}\left(a,b,p\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
b:Scale parameterb:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}