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Pareto distribution

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Probability distribution

Pareto Type I
Probability density function
Pareto Type I probability density functions for various α
Pareto Type I probability density functions for variousα{\displaystyle \alpha } withxm=1.{\displaystyle x_{\mathrm {m} }=1.} Asα→∞,{\displaystyle \alpha \rightarrow \infty ,} the distribution approachesδ(x−xm),{\displaystyle \delta (x-x_{\mathrm {m} }),} whereδ{\displaystyle \delta } is theDirac delta function.
Cumulative distribution function
Pareto Type I cumulative distribution functions for various α
Pareto Type I cumulative distribution functions for variousα{\displaystyle \alpha } withxm=1.{\displaystyle x_{\mathrm {m} }=1.}
Parametersxm>0{\displaystyle x_{\mathrm {m} }>0}scale (real)
α>0{\displaystyle \alpha >0}shape (real)
Supportx∈[xm,∞){\displaystyle x\in [x_{\mathrm {m} },\infty )}
PDFαxmαxα+1{\displaystyle {\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}}
CDF1−(xmx)α{\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }}
Quantilexm(1−p)−1α{\displaystyle x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}}
Mean{∞for α≤1αxmα−1for α>1{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}
Medianxm2α{\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}
Modexm{\displaystyle x_{\mathrm {m} }}
Variance{∞for α≤2xm2α(α−1)2(α−2)for α>2{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 2\\{\dfrac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}}
Skewness2(1+α)α−3α−2α for α>3{\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}
Excess kurtosis6(α3+α2−6α−2)α(α−3)(α−4) for α>4{\displaystyle {\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4}
Entropylog⁡((xmα)e1+1α){\displaystyle \log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)}
MGFdoes not exist
CFα(−ixmt)αΓ(−α,−ixmt){\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}
Fisher informationI(xm,α)=[α2xm2001α2]{\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}
Expected shortfallxmα(1−p)1α(α−1){\displaystyle {\frac {x_{m}\alpha }{(1-p)^{\frac {1}{\alpha }}(\alpha -1)}}}[1]

ThePareto distribution, named after the Italiancivil engineer,economist, andsociologistVilfredo Pareto,[2] is apower-lawprobability distribution that is used in description ofsocial,quality control,scientific,geophysical,actuarial, and many other types of observable phenomena; the principle originally applied to describing thedistribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.[3][4]

ThePareto principle or "80:20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value (α)of  log 4 5 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80:20 distribution fits a wide range of cases, including natural phenomena[5] and human activities.[6][7]

Definitions

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IfX is arandom variable with a Pareto (Type I) distribution,[8] then the probability thatX is greater than some numberx, i.e., thesurvival function (also called tail function), is given by

F¯(x)=Pr(X>x)={(xmx)αx≥xm,1x<xm,{\displaystyle {\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\1&x<x_{\mathrm {m} },\end{cases}}}

wherexm is the (necessarily positive) minimum possible value ofX, andα is a positive parameter. The type I Pareto distribution is characterized by ascale parameterxm and ashape parameterα, which is known as thetail index. If this distribution is used to model the distribution of wealth, then the parameterα is called thePareto index.

Cumulative distribution function

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From the definition, thecumulative distribution function of a Pareto random variable with parametersα andxm is

FX(x)={1−(xmx)αx≥xm,0x<xm.{\displaystyle F_{X}(x)={\begin{cases}1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}

Probability density function

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It follows (bydifferentiation) that theprobability density function is

fX(x)={αxmαxα+1x≥xm,0x<xm.{\displaystyle f_{X}(x)={\begin{cases}{\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}}

When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axesasymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in alog–log plot, the distribution is represented by a straight line.

Properties

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Moments and characteristic function

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The parameters may be solved for using themethod of moments.[9]

Conditional distributions

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Theconditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number x1{\displaystyle x_{1}} exceedingxm{\displaystyle x_{\text{m}}}, is a Pareto distribution with the same Pareto index α{\displaystyle \alpha } but with minimum x1{\displaystyle x_{1}} instead ofxm{\displaystyle x_{\text{m}}}:

Pr(X≥x|X≥x1)={(x1x)αx≥x1,1x<x1.{\displaystyle {\text{Pr}}(X\geq x|X\geq x_{1})={\begin{cases}\left({\frac {x_{1}}{x}}\right)^{\alpha }&x\geq x_{1},\\1&x<x_{1}.\end{cases}}}

This implies that the conditional expected value (if it is finite, i.e.α>1{\displaystyle \alpha >1}) is proportional tox1{\displaystyle x_{1}}:

E(X|X≥x1)∝x1.{\displaystyle {\text{E}}(X|X\geq x_{1})\propto x_{1}.}

In case of random variables that describe the lifetime of an object, this means that life expectancy is proportional to age, and is called theLindy effect or Lindy's Law.[10]

A characterization theorem

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SupposeX1,X2,X3,…{\displaystyle X_{1},X_{2},X_{3},\dotsc } areindependent identically distributedrandom variables whose probability distribution is supported on the interval[xm,∞){\displaystyle [x_{\text{m}},\infty )} for somexm>0{\displaystyle x_{\text{m}}>0}. Suppose that for alln{\displaystyle n}, the two random variablesmin{X1,…,Xn}{\displaystyle \min\{X_{1},\dotsc ,X_{n}\}} and(X1+⋯+Xn)/min{X1,…,Xn}{\displaystyle (X_{1}+\dotsb +X_{n})/\min\{X_{1},\dotsc ,X_{n}\}} are independent. Then the common distribution is a Pareto distribution.[citation needed]

Geometric mean

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Thegeometric mean (G) is[11]

G=xmexp⁡(1α).{\displaystyle G=x_{\text{m}}\exp \left({\frac {1}{\alpha }}\right).}

Harmonic mean

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Theharmonic mean (H) is[11]

H=xm(1+1α).{\displaystyle H=x_{\text{m}}\left(1+{\frac {1}{\alpha }}\right).}

Graphical representation

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The characteristic curved 'long tail' distribution, when plotted on a linear scale, masks the underlying simplicity of the function when plotted on alog-log graph, which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that forx ≥xm,

log⁡fX(x)=log⁡(αxmαxα+1)=log⁡(αxmα)−(α+1)log⁡x.{\displaystyle \log f_{X}(x)=\log \left(\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}\right)=\log(\alpha x_{\mathrm {m} }^{\alpha })-(\alpha +1)\log x.}

Sinceα is positive, the gradient −(α + 1) is negative.

Related distributions

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Generalized Pareto distributions

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See also:Generalized Pareto distribution

There is a hierarchy[8][12] of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.[8][12][13] Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto[12][14] distribution generalizes Pareto Type IV.

Pareto types I–IV

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The Pareto distribution hierarchy is summarized in the next table comparing thesurvival functions (complementary CDF).

Whenμ = 0, the Pareto distribution Type II is also known as theLomax distribution.[15]

In this section, the symbolxm, used before to indicate the minimum value ofx, is replaced by σ.

Pareto distributions
F¯(x)=1−F(x){\displaystyle {\overline {F}}(x)=1-F(x)}SupportParameters
Type I[xσ]−α{\displaystyle \left[{\frac {x}{\sigma }}\right]^{-\alpha }}x≥σ{\displaystyle x\geq \sigma }σ>0,α{\displaystyle \sigma >0,\alpha }
Type II[1+x−μσ]−α{\displaystyle \left[1+{\frac {x-\mu }{\sigma }}\right]^{-\alpha }}x≥μ{\displaystyle x\geq \mu }μ∈R,σ>0,α{\displaystyle \mu \in \mathbb {R} ,\sigma >0,\alpha }
Lomax[1+xσ]−α{\displaystyle \left[1+{\frac {x}{\sigma }}\right]^{-\alpha }}x≥0{\displaystyle x\geq 0}σ>0,α{\displaystyle \sigma >0,\alpha }
Type III[1+(x−μσ)1/γ]−1{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-1}}x≥μ{\displaystyle x\geq \mu }μ∈R,σ,γ>0{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0}
Type IV[1+(x−μσ)1/γ]−α{\displaystyle \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-\alpha }}x≥μ{\displaystyle x\geq \mu }μ∈R,σ,γ>0,α{\displaystyle \mu \in \mathbb {R} ,\sigma ,\gamma >0,\alpha }

The shape parameterα is thetail index,μ is location,σ is scale,γ is an inequality parameter. Some special cases of Pareto Type (IV) are

P(IV)(σ,σ,1,α)=P(I)(σ,α),{\displaystyle P(IV)(\sigma ,\sigma ,1,\alpha )=P(I)(\sigma ,\alpha ),}P(IV)(μ,σ,1,α)=P(II)(μ,σ,α),{\displaystyle P(IV)(\mu ,\sigma ,1,\alpha )=P(II)(\mu ,\sigma ,\alpha ),}P(IV)(μ,σ,γ,1)=P(III)(μ,σ,γ).{\displaystyle P(IV)(\mu ,\sigma ,\gamma ,1)=P(III)(\mu ,\sigma ,\gamma ).}

The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail indexα (inequality indexγ). In particular, fractionalδ-moments are finite for someδ > 0, as shown in the table below, whereδ is not necessarily an integer.

Moments of Pareto I–IV distributions (caseμ = 0)
E⁡[X]{\displaystyle \operatorname {E} [X]}ConditionE⁡[Xδ]{\displaystyle \operatorname {E} [X^{\delta }]}Condition
Type Iσαα−1{\displaystyle {\frac {\sigma \alpha }{\alpha -1}}}α>1{\displaystyle \alpha >1}σδαα−δ{\displaystyle {\frac {\sigma ^{\delta }\alpha }{\alpha -\delta }}}δ<α{\displaystyle \delta <\alpha }
Type IIσα−1+μ{\displaystyle {\frac {\sigma }{\alpha -1}}+\mu }α>1{\displaystyle \alpha >1}σδΓ(α−δ)Γ(1+δ)Γ(α){\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\delta )\Gamma (1+\delta )}{\Gamma (\alpha )}}}0<δ<α,μ=0{\displaystyle 0<\delta <\alpha ,\mu =0}
Type IIIσΓ(1−γ)Γ(1+γ){\displaystyle \sigma \Gamma (1-\gamma )\Gamma (1+\gamma )}−1<γ<1{\displaystyle -1<\gamma <1}σδΓ(1−γδ)Γ(1+γδ){\displaystyle \sigma ^{\delta }\Gamma (1-\gamma \delta )\Gamma (1+\gamma \delta )}−γ−1<δ<γ−1{\displaystyle -\gamma ^{-1}<\delta <\gamma ^{-1}}
Type IVσΓ(α−γ)Γ(1+γ)Γ(α){\displaystyle {\frac {\sigma \Gamma (\alpha -\gamma )\Gamma (1+\gamma )}{\Gamma (\alpha )}}}−1<γ<α{\displaystyle -1<\gamma <\alpha }σδΓ(α−γδ)Γ(1+γδ)Γ(α){\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\gamma \delta )\Gamma (1+\gamma \delta )}{\Gamma (\alpha )}}}−γ−1<δ<α/γ{\displaystyle -\gamma ^{-1}<\delta <\alpha /\gamma }

Feller–Pareto distribution

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Feller[12][14] defines a Pareto variable by transformationU = Y−1 − 1 of abeta random variable ,Y, whose probability density function is

f(y)=yγ1−1(1−y)γ2−1B(γ1,γ2),0<y<1;γ1,γ2>0,{\displaystyle f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 0<y<1;\gamma _{1},\gamma _{2}>0,}

where B( ) is thebeta function. If

W=μ+σ(Y−1−1)γ,σ>0,γ>0,{\displaystyle W=\mu +\sigma (Y^{-1}-1)^{\gamma },\qquad \sigma >0,\gamma >0,}

thenW has a Feller–Pareto distribution FP(μ,σ,γ,γ1,γ2).[8]

IfU1∼Γ(δ1,1){\displaystyle U_{1}\sim \Gamma (\delta _{1},1)} andU2∼Γ(δ2,1){\displaystyle U_{2}\sim \Gamma (\delta _{2},1)} are independentGamma variables, another construction of a Feller–Pareto (FP) variable is[16]

W=μ+σ(U1U2)γ{\displaystyle W=\mu +\sigma \left({\frac {U_{1}}{U_{2}}}\right)^{\gamma }}

and we writeW ~ FP(μ,σ,γ,δ1,δ2). Special cases of the Feller–Pareto distribution are

FP(σ,σ,1,1,α)=P(I)(σ,α){\displaystyle FP(\sigma ,\sigma ,1,1,\alpha )=P(I)(\sigma ,\alpha )}FP(μ,σ,1,1,α)=P(II)(μ,σ,α){\displaystyle FP(\mu ,\sigma ,1,1,\alpha )=P(II)(\mu ,\sigma ,\alpha )}FP(μ,σ,γ,1,1)=P(III)(μ,σ,γ){\displaystyle FP(\mu ,\sigma ,\gamma ,1,1)=P(III)(\mu ,\sigma ,\gamma )}FP(μ,σ,γ,1,α)=P(IV)(μ,σ,γ,α).{\displaystyle FP(\mu ,\sigma ,\gamma ,1,\alpha )=P(IV)(\mu ,\sigma ,\gamma ,\alpha ).}

Inverse-Pareto Distribution / Power Distribution

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When a random variableY{\displaystyle Y} follows a pareto distribution, then its inverseX=1/Y{\displaystyle X=1/Y} follows a Power distribution. Inverse Pareto distribution is equivalent to a Power distribution[17]

Y∼Pa(α,xm)=αxmαyα+1(y≥xm)⇔X∼iPa(α,xm)=Power(xm−1,α)=αxα−1(xm−1)α(0<x≤xm−1){\displaystyle Y\sim \mathrm {Pa} (\alpha ,x_{m})={\frac {\alpha x_{m}^{\alpha }}{y^{\alpha +1}}}\quad (y\geq x_{m})\quad \Leftrightarrow \quad X\sim \mathrm {iPa} (\alpha ,x_{m})=\mathrm {Power} (x_{m}^{-1},\alpha )={\frac {\alpha x^{\alpha -1}}{(x_{m}^{-1})^{\alpha }}}\quad (0<x\leq x_{m}^{-1})}

Relation to the exponential distribution

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The Pareto distribution is related to theexponential distribution as follows. IfX is Pareto-distributed with minimumxm and index α, then

Y=log⁡(Xxm){\displaystyle Y=\log \left({\frac {X}{x_{\mathrm {m} }}}\right)}

isexponentially distributed with rate parameter α. Equivalently, ifY is exponentially distributed with rate α, then

xmeY{\displaystyle x_{\mathrm {m} }e^{Y}}

is Pareto-distributed with minimumxm and index α.

This can be shown using the standard change-of-variable techniques:

Pr(Y<y)=Pr(log⁡(Xxm)<y)=Pr(X<xmey)=1−(xmxmey)α=1−e−αy.{\displaystyle {\begin{aligned}\Pr(Y<y)&=\Pr \left(\log \left({\frac {X}{x_{\mathrm {m} }}}\right)<y\right)\\&=\Pr(X<x_{\mathrm {m} }e^{y})=1-\left({\frac {x_{\mathrm {m} }}{x_{\mathrm {m} }e^{y}}}\right)^{\alpha }=1-e^{-\alpha y}.\end{aligned}}}

The last expression is the cumulative distribution function of an exponential distribution with rate α.

Pareto distribution can be constructed by hierarchical exponential distributions.[18] Letϕ|a∼Exp(a){\displaystyle \phi |a\sim {\text{Exp}}(a)} andη|ϕ∼Exp(ϕ){\displaystyle \eta |\phi \sim {\text{Exp}}(\phi )}. Then we havep(η|a)=a(a+η)2{\displaystyle p(\eta |a)={\frac {a}{(a+\eta )^{2}}}} and, as a result,a+η∼Pareto(a,1){\displaystyle a+\eta \sim {\text{Pareto}}(a,1)}.

More in general, ifλ∼Gamma(α,β){\displaystyle \lambda \sim {\text{Gamma}}(\alpha ,\beta )} (shape-rate parametrization) andη|λ∼Exp(λ){\displaystyle \eta |\lambda \sim {\text{Exp}}(\lambda )}, thenβ+η∼Pareto(β,α){\displaystyle \beta +\eta \sim {\text{Pareto}}(\beta ,\alpha )}.

Equivalently, ifY∼Gamma(α,1){\displaystyle Y\sim {\text{Gamma}}(\alpha ,1)} andX∼Exp(1){\displaystyle X\sim {\text{Exp}}(1)}, thenxm(1+XY)∼Pareto(xm,α){\displaystyle x_{\text{m}}\!\left(1+{\frac {X}{Y}}\right)\sim {\text{Pareto}}(x_{\text{m}},\alpha )}.

Relation to the log-normal distribution

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The Pareto distribution andlog-normal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively theexponential distribution andnormal distribution. (Seethe previous section.)

Relation to the generalized Pareto distribution

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The Pareto distribution is a special case of thegeneralized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with theLomax distribution as a special case. This family also contains both the unshifted and shiftedexponential distributions.

The Pareto distribution with scalexm{\displaystyle x_{m}} and shapeα{\displaystyle \alpha } is equivalent to the generalized Pareto distribution with locationμ=xm{\displaystyle \mu =x_{m}}, scaleσ=xm/α{\displaystyle \sigma =x_{m}/\alpha } and shapeξ=1/α{\displaystyle \xi =1/\alpha } and, conversely, one can get the Pareto distribution from the GPD by takingxm=σ/ξ{\displaystyle x_{m}=\sigma /\xi } andα=1/ξ{\displaystyle \alpha =1/\xi } ifξ>0{\displaystyle \xi >0}.

Bounded Pareto distribution

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See also:Truncated distribution
Bounded Pareto
Parameters

L>0{\displaystyle L>0}location (real)
H>L{\displaystyle H>L}location (real)

α>0{\displaystyle \alpha >0}shape (real)
SupportL⩽x⩽H{\displaystyle L\leqslant x\leqslant H}
PDFαLαx−α−11−(LH)α{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}
CDF1−Lαx−α1−(LH)α{\displaystyle {\frac {1-L^{\alpha }x^{-\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}
Mean{Lα1−(LH)α(αα−1)(1Lα−1−1Hα−1),α≠1HLH−Lln⁡HL,α=1{\displaystyle {\begin{cases}{\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\left({\frac {\alpha }{\alpha -1}}\right)\left({\frac {1}{L^{\alpha -1}}}-{\frac {1}{H^{\alpha -1}}}\right),&\alpha \neq 1\\{\frac {{H}{L}}{{H}-{L}}}\ln {\frac {H}{L}},&\alpha =1\end{cases}}}
MedianL(1−12(1−(LH)α))−1α{\displaystyle L\left(1-{\frac {1}{2}}\left(1-\left({\frac {L}{H}}\right)^{\alpha }\right)\right)^{-{\frac {1}{\alpha }}}}
Variance

{Lα1−(LH)α(αα−2)(1Lα−2−1Hα−2),α≠22H2L2H2−L2ln⁡HL,α=2{\displaystyle {\begin{cases}{\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\left({\frac {\alpha }{\alpha -2}}\right)\left({\frac {1}{L^{\alpha -2}}}-{\frac {1}{H^{\alpha -2}}}\right),&\alpha \neq 2\\{\frac {2{H}^{2}{L}^{2}}{{H}^{2}-{L}^{2}}}\ln {\frac {H}{L}},&\alpha =2\end{cases}}}

(this is the second raw moment, not the variance)
Skewness

Lα1−(LH)α⋅α(Lk−α−Hk−α)(α−k),α≠j{\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot {\frac {\alpha (L^{k-\alpha }-H^{k-\alpha })}{(\alpha -k)}},\alpha \neq j}

(this is the kth raw moment, not the skewness)

The bounded (or truncated) Pareto distribution has three parameters:α,L andH. As in the standard Pareto distributionα determines the shape.L denotes the minimal value, andH denotes the maximal value.

Theprobability density function is

αLαx−α−11−(LH)α,{\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}},}

whereL ≤ x ≤ H, andα > 0.

Generating bounded Pareto random variables

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IfU isuniformly distributed on (0, 1), then applying inverse-transform method[19]

U=1−Lαx−α1−(LH)α{\displaystyle U={\frac {1-L^{\alpha }x^{-\alpha }}{1-({\frac {L}{H}})^{\alpha }}}}x=(−UHα−ULα−HαHαLα)−1α{\displaystyle x=\left(-{\frac {UH^{\alpha }-UL^{\alpha }-H^{\alpha }}{H^{\alpha }L^{\alpha }}}\right)^{-{\frac {1}{\alpha }}}}

is a bounded Pareto-distributed.

Symmetric Pareto distribution

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The purpose of the Symmetric and Zero Symmetric Pareto distributions is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from the Pareto distribution. Long probability tails normally means that probability decays slowly, and can be used to fit a variety of datasets. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.[20]

The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:[20]

F(X)=P(x<X)={12(b2b−X)aX<b1−12(bX)aX≥b{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over 2b-X})^{a}&X<b\\1-{\tfrac {1}{2}}({\tfrac {b}{X}})^{a}&X\geq b\end{cases}}}

The corresponding probability density function (PDF) is:[20]

p(x)=aba2(b+|x−b|)a+1,X∈R{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x-b\right\vert )^{a+1}},X\in R}

This distribution has two parameters: a and b. It is symmetric about b. Then the mathematic expectation is b. When, it has variance as following:

E((x−b)2)=∫−∞∞(x−b)2p(x)dx=2b2(a−2)(a−1){\displaystyle E((x-b)^{2})=\int _{-\infty }^{\infty }(x-b)^{2}p(x)dx={\frac {2b^{2}}{(a-2)(a-1)}}}

The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:

F(X)=P(x<X)={12(bb−X)aX<01−12(bb+X)aX≥0{\displaystyle F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over b-X})^{a}&X<0\\1-{\tfrac {1}{2}}({\tfrac {b}{b+X}})^{a}&X\geq 0\end{cases}}}

The corresponding PDF is:

p(x)=aba2(b+|x|)a+1,X∈R{\displaystyle p(x)={ab^{a} \over 2(b+\left\vert x\right\vert )^{a+1}},X\in R}

This distribution is symmetric about zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.[20]

Multivariate Pareto distribution

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The univariate Pareto distribution has been extended to amultivariate Pareto distribution.[21]

Statistical inference

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Estimation of parameters

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Thelikelihood function for the Pareto distribution parametersα andxm, given an independentsamplex = (x1, x2, ..., xn), is

L(α,xm)=∏i=1nαxmαxiα+1=αnxmnα∏i=1n1xiα+1.{\displaystyle L(\alpha ,x_{\mathrm {m} })=\prod _{i=1}^{n}\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x_{i}^{\alpha +1}}}=\alpha ^{n}x_{\mathrm {m} }^{n\alpha }\prod _{i=1}^{n}{\frac {1}{x_{i}^{\alpha +1}}}.}

Therefore, the logarithmic likelihood function is

ℓ(α,xm)=nln⁡α+nαln⁡xm−(α+1)∑i=1nln⁡xi.{\displaystyle \ell (\alpha ,x_{\mathrm {m} })=n\ln \alpha +n\alpha \ln x_{\mathrm {m} }-(\alpha +1)\sum _{i=1}^{n}\ln x_{i}.}

It can be seen thatℓ(α,xm){\displaystyle \ell (\alpha ,x_{\mathrm {m} })} is monotonically increasing withxm, that is, the greater the value ofxm, the greater the value of the likelihood function. Hence, sincex ≥xm, we conclude that

x^m=minixi.{\displaystyle {\widehat {x}}_{\mathrm {m} }=\min _{i}{x_{i}}.}

To find theestimator forα, we compute the corresponding partial derivative and determine where it is zero:

∂ℓ∂α=nα+nln⁡xm−∑i=1nln⁡xi=0.{\displaystyle {\frac {\partial \ell }{\partial \alpha }}={\frac {n}{\alpha }}+n\ln x_{\mathrm {m} }-\sum _{i=1}^{n}\ln x_{i}=0.}

Thus themaximum likelihood estimator forα is:

α^=n∑iln⁡(xi/x^m).{\displaystyle {\widehat {\alpha }}={\frac {n}{\sum _{i}\ln(x_{i}/{\widehat {x}}_{\mathrm {m} })}}.}

The expected statistical error is:[22]

σ=α^n.{\displaystyle \sigma ={\frac {\widehat {\alpha }}{\sqrt {n}}}.}

Malik (1970)[23] gives the exact joint distribution of(x^m,α^){\displaystyle ({\hat {x}}_{\mathrm {m} },{\hat {\alpha }})}. In particular,x^m{\displaystyle {\hat {x}}_{\mathrm {m} }} andα^{\displaystyle {\hat {\alpha }}} areindependent andx^m{\displaystyle {\hat {x}}_{\mathrm {m} }} is Pareto with scale parameterxm and shape parameternα, whereasα^{\displaystyle {\hat {\alpha }}} has aninverse-gamma distribution with shape and scale parametersn − 1 andnα, respectively.

Occurrence and applications

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General

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Vilfredo Pareto originally used this distribution to describe theallocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.[4] This idea is sometimes expressed more simply as thePareto principle or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.[24] As Michael Hudson points out inThe Collapse of Antiquity, "a mathematical corollary [is] that 10% would have 65% of the wealth, and 5% would have half the national wealth."[25] However, the 80-20 rule corresponds to a particular value ofα, and in fact, Pareto's data on British income taxes in hisCours d'économie politique indicates that about 30% of the population had about 70% of the income.[citation needed] Theprobability density function (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact,net worth may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of size or magnitude. The following examples are sometimes seen as approximately Pareto-distributed:

  • All four variables of household budget constraints: consumption, labor income, capital income, and wealth.[26]
  • The sizes of human settlements (a few large cities, many hamlets/villages)[27][28]
  • File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)[27]
  • Hard disk drive error rates[29]
  • Clusters ofBose–Einstein condensate nearabsolute zero[30]
Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using CumFreq, see alsodistribution fitting

Relation to Zipf's law

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The Pareto distribution is a continuous probability distribution.Zipf's law, also sometimes called thezeta distribution, is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if thex{\displaystyle x} values (incomes) are binned intoN{\displaystyle N} ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by definingxm{\displaystyle x_{m}} so thatαxmα=1H(N,α−1){\displaystyle \alpha x_{\mathrm {m} }^{\alpha }={\frac {1}{H(N,\alpha -1)}}} whereH(N,α−1){\displaystyle H(N,\alpha -1)} is thegeneralized harmonic number. This makes Zipf's probability density function derivable from Pareto's.

f(x)=αxmαxα+1=1xsH(N,s){\displaystyle f(x)={\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}={\frac {1}{x^{s}H(N,s)}}}

wheres=α−1{\displaystyle s=\alpha -1} andx{\displaystyle x} is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) hasf(x){\displaystyle f(x)} probability of rankingx{\displaystyle x}.

Relation to the "Pareto principle"

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The "80/20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index isα=log4⁡5=log10⁡5log10⁡4≈1.161{\displaystyle \alpha =\log _{4}5={\cfrac {\log _{10}5}{\log _{10}4}}\approx 1.161}. This result can be derived from theLorenz curve formula given below. Moreover, the following have been shown[35] to be mathematically equivalent:

This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.

This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have an infinite expected value, and so cannot reasonably model income distribution.

Relation to Price's law

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Price's law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case thatα=1{\displaystyle \alpha =1}. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.

Lorenz curve and Gini coefficient

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Lorenz curves for a number of Pareto distributions. The caseα = ∞ corresponds to perfectly equal distribution (G = 0) and theα = 1 line corresponds to complete inequality (G = 1)

TheLorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curveL(F) is written in terms of the PDFf or the CDFF as

L(F)=∫xmx(F)xf(x)dx∫xm∞xf(x)dx=∫0Fx(F′)dF′∫01x(F′)dF′{\displaystyle L(F)={\frac {\int _{x_{\mathrm {m} }}^{x(F)}xf(x)\,dx}{\int _{x_{\mathrm {m} }}^{\infty }xf(x)\,dx}}={\frac {\int _{0}^{F}x(F')\,dF'}{\int _{0}^{1}x(F')\,dF'}}}

wherex(F) is the inverse of the CDF. For the Pareto distribution,

x(F)=xm(1−F)1α{\displaystyle x(F)={\frac {x_{\mathrm {m} }}{(1-F)^{\frac {1}{\alpha }}}}}

and the Lorenz curve is calculated to be

L(F)=1−(1−F)1−1α,{\displaystyle L(F)=1-(1-F)^{1-{\frac {1}{\alpha }}},}

For0<α≤1{\displaystyle 0<\alpha \leq 1} the denominator is infinite, yieldingL=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.

According toOxfam (2016) the richest 62 people have as much wealth as the poorest half of the world's population.[36] We can estimate the Pareto index that would apply to this situation. Letting ε equal62/(7×109){\displaystyle 62/(7\times 10^{9})} we have:L(1/2)=1−L(1−ε){\displaystyle L(1/2)=1-L(1-\varepsilon )}or1−(1/2)1−1α=ε1−1α{\displaystyle 1-(1/2)^{1-{\frac {1}{\alpha }}}=\varepsilon ^{1-{\frac {1}{\alpha }}}}The solution is thatα equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.[37]

TheGini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (α = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (forα≥1{\displaystyle \alpha \geq 1}) to be

G=1−2(∫01L(F)dF)=12α−1{\displaystyle G=1-2\left(\int _{0}^{1}L(F)\,dF\right)={\frac {1}{2\alpha -1}}}

(see Aaberge 2005).

Random variate generation

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Further information:Non-uniform random variate generation

Random samples can be generated usinginverse transform sampling. Given a random variateU drawn from theuniform distribution on the unit interval [0, 1], the variateT given by

T=xmU1/α{\displaystyle T={\frac {x_{\mathrm {m} }}{U^{1/\alpha }}}}

is Pareto-distributed.[38]

See also

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References

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  1. ^abNorton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019)."Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"(PDF).Annals of Operations Research.299 (1–2). Springer:1281–1315.arXiv:1811.11301.doi:10.1007/s10479-019-03373-1.S2CID 254231768. Archived fromthe original(PDF) on 2023-03-31. Retrieved2023-02-27.
  2. ^Amoroso, Luigi (January 1938). "Vilfredo Pareto".Econometrica (Pre-1986).6 (1).
  3. ^Pareto, Vilfredo (1898)."Cours d'economie politique".Journal of Political Economy.6.doi:10.1086/250536.
  4. ^abPareto, Vilfredo,Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, 1964, pp. 299–345.Original book archived
  5. ^van Montfort, M.A.J. (1986)."The generalized Pareto distribution applied to rainfall depths".Hydrological Sciences Journal.31 (2):151–162.Bibcode:1986HydSJ..31..151V.doi:10.1080/02626668609491037.
  6. ^Oancea, Bogdan (2017). "Income inequality in Romania: The exponential-Pareto distribution".Physica A: Statistical Mechanics and Its Applications.469:486–498.Bibcode:2017PhyA..469..486O.doi:10.1016/j.physa.2016.11.094.
  7. ^Morella, Matteo."Pareto distribution".academia.edu.
  8. ^abcdArnold, Barry C. (1983).Pareto Distributions. International Co-operative Publishing House.ISBN 978-0-89974-012-6.
  9. ^S. Hussain, S.H. Bhatti (2018).Parameter estimation of Pareto distribution: Some modified moment estimators.Maejo International Journal of Science and Technology 12(1):11-27.
  10. ^Eliazar, Iddo (November 2017). "Lindy's Law".Physica A: Statistical Mechanics and Its Applications.486:797–805.Bibcode:2017PhyA..486..797E.doi:10.1016/j.physa.2017.05.077.S2CID 125349686.
  11. ^abJohnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.
  12. ^abcdJohnson, Kotz, and Balakrishnan (1994), (20.4).
  13. ^Christian Kleiber & Samuel Kotz (2003).Statistical Size Distributions in Economics and Actuarial Sciences.Wiley.ISBN 978-0-471-15064-0.
  14. ^abFeller, W. (1971).An Introduction to Probability Theory and its Applications. Vol. II (2nd ed.). New York: Wiley. p. 50. "The densities (4.3) are sometimes called after the economistPareto. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~Ax−α asx → ∞".
  15. ^Lomax, K. S. (1954). "Business failures. Another example of the analysis of failure data".Journal of the American Statistical Association.49 (268):847–52.doi:10.1080/01621459.1954.10501239.
  16. ^Chotikapanich, Duangkamon (16 September 2008)."Chapter 7: Pareto and Generalized Pareto Distributions".Modeling Income Distributions and Lorenz Curves. Springer. pp. 121–22.ISBN 9780387727967.
  17. ^Dallas, A. C. (1976). "Characterizing the Pareto and power distributions".Annals of the Institute of Statistical Mathematics.28 (1):491–497.doi:10.1007/BF02504764.
  18. ^White, Gentry (2006).Bayesian semiparametric spatial and joint spatio-temporal modeling (Phd thesis). University of Missouri–Columbia.hdl:10355/4450. section 5.3.1.
  19. ^"Inverse Transform Method". Archived fromthe original on 2012-01-17. Retrieved2012-08-27.
  20. ^abcdHuang, Xiao-dong (2004). "A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic".IEEE Transactions on Broadcasting.50 (3):323–334.Bibcode:2004ITBE...50..323H.doi:10.1109/TBC.2004.834013.
  21. ^Rootzén, Holger; Tajvidi, Nader (2006). "Multivariate generalized Pareto distributions".Bernoulli.12 (5):917–30.CiteSeerX 10.1.1.145.2991.doi:10.3150/bj/1161614952.S2CID 16504396.
  22. ^M. E. J. Newman (2005). "Power laws, Pareto distributions and Zipf's law".Contemporary Physics.46 (5):323–51.arXiv:cond-mat/0412004.Bibcode:2005ConPh..46..323N.doi:10.1080/00107510500052444.S2CID 202719165.
  23. ^H. J. Malik (1970). "Estimation of the Parameters of the Pareto Distribution".Metrika.15:126–132.doi:10.1007/BF02613565.S2CID 124007966.
  24. ^For a two-quantile population, where approximately 18% of the population owns 82% of the wealth, theTheil index takes the value 1.
  25. ^Hudson, Michael (2023).The Collapse of Antiquity. p. 85, n. 7.
  26. ^Gaillard, Alexandre; Hellwig, Christian; Wangner, Philipp; Werquin, Nicolas (2023)."Consumption, Wealth, and Income Inequality: A Tale of Tails".doi:10.2139/ssrn.4636704.SSRN 4636704.
  27. ^abcdeReed, William J.; et al. (2004). "The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions".Communications in Statistics – Theory and Methods.33 (8):1733–53.CiteSeerX 10.1.1.70.4555.doi:10.1081/sta-120037438.S2CID 13906086.
  28. ^Reed, William J. (2002). "On the rank-size distribution for human settlements".Journal of Regional Science.42 (1):1–17.Bibcode:2002JRegS..42....1R.doi:10.1111/1467-9787.00247.S2CID 154285730.
  29. ^Schroeder, Bianca; Damouras, Sotirios; Gill, Phillipa (2010-02-24)."Understanding latent sector error and how to protect against them"(PDF).8th Usenix Conference on File and Storage Technologies (FAST 2010). Retrieved2010-09-10.We experimented with 5 different distributions (Geometric, Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ2 statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.
  30. ^Yuji Ijiri; Simon, Herbert A. (May 1975)."Some Distributions Associated with Bose–Einstein Statistics".Proc. Natl. Acad. Sci. USA.72 (5):1654–57.Bibcode:1975PNAS...72.1654I.doi:10.1073/pnas.72.5.1654.PMC 432601.PMID 16578724.
  31. ^Harchol-Balter, Mor; Downey, Allen (August 1997)."Exploiting Process Lifetime Distributions for Dynamic Load Balancing"(PDF).ACM Transactions on Computer Systems.15 (3):253–258.doi:10.1145/263326.263344.S2CID 52861447.
  32. ^Kleiber and Kotz (2003): p. 94.
  33. ^Seal, H. (1980)."Survival probabilities based on Pareto claim distributions".ASTIN Bulletin.11:61–71.doi:10.1017/S0515036100006620.
  34. ^CumFreq, software for cumulative frequency analysis and probability distribution fitting[1]
  35. ^Hardy, Michael (2010). "Pareto's Law".Mathematical Intelligencer.32 (3):38–43.doi:10.1007/s00283-010-9159-2.S2CID 121797873.
  36. ^"62 people own the same as half the world, reveals Oxfam Davos report". Oxfam. Jan 2016.
  37. ^"Global Wealth Report 2013". Credit Suisse. Oct 2013. p. 22. Archived fromthe original on 2015-02-14. Retrieved2016-01-24.
  38. ^Tanizaki, Hisashi (2004).Computational Methods in Statistics and Econometrics. CRC Press. p. 133.ISBN 9780824750886.

Notes

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