The Basic Pareto Distribution - Random Services

The Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model the distribution of incomes and other financial variables. \(\newcommand{\P}{\mathbb{P}}\) \(\newcommand{\E}{\mathbb{E}}\) \(\newcommand{\var}{\text{var}}\) \(\newcommand{\sd}{\text{sd}}\) \(\newcommand{\N}{\mathbb{N}}\) \(\newcommand{\skw}{\text{skew}}\) \(\newcommand{\kur}{\text{kurt}}\) TheParetodistributionisaskewed,heavy-taileddistributionthatissometimesusedtomodelthedistributionofincomesandotherfinancialvariables. TheBasicParetoDistribution DistributionFunctions ThebasicParetodistributionwithshapeparameter\(a\in(0,\infty)\)isacontinuousdistributionon\([1,\infty)\)withdistributionfunction\(G\)givenby \[G(z)=1-\frac{1}{z^a},\quadz\in[1,\infty)\] Thespecialcase\(a=1\)givesthestandardParetodistribuiton. Proof: Clearly\(G\)isincreasingandcontinuouson\([1,\infty)\),with\(G(1)=0\)and\(G(z)\to1\)as\(z\to\infty\). TheParetodistributionisnamedfortheeconomistVilfredoPareto. Theprobabilitydensityfunction\(g\)isgivenby \[g(z)=\frac{a}{z^{a+1}},\quadz\in[1,\infty)\] \(g\)isdecreasingwithmode\(z=1\) \(g\)isconcaveupward. Proof: Recallthat\(g=G^\prime\).Parts(a)and(b)followfromstandardcalculus. ThereasonthattheParetodistributionisheavy-tailedisthatthe\(g\)decreasesatapowerrateratherthananexponentialrate. OpenthespecialdistributionsimulatorandselecttheParetodistribution.Varytheshapeparameterandnotetheshapeoftheprobabilitydensityfunction.Forselectedvaluesoftheparameter,runthesimulation1000timesandcomparetheempiricaldensityfunctiontotheprobabilitydensityfunction. Thequantilefunction\(G^{-1}\)isgivenby \[G^{-1}(p)=\frac{1}{(1-p)^{1/a}},\quadp\in[0,1)\] Thefirstquartileis\(q_1=\left(\frac{4}{3}\right)^{1/a}\). Themedianis\(q_2=2^{1/a}\). Thethirdquartileis\(q_3=4^{1/a}\). Proof: Theformulafor\(G^{-1}(p)\)comesfromsolving\(G(z)=p\)for\(z\)intermsof\(p\). OpenthespecialdistributioncalculatorandselecttheParetodistribution.Varytheshapeparameterandnotetheshapeoftheprobabilitydensityanddistributionfunctions.Forselectedvaluesoftheparameters,computeafewvaluesofthedistributionandquantilefunctions. Moments Supposethatrandomvariable\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\in(0,\infty)\).Becausethedistributionisheavy-tailed,themean,variance,andothermomentsof\(Z\)arefiniteonlyiftheshapeparameter\(a\)issufficientlylarge. Themomentsof\(Z\)(about0)are \(\E(Z^n)=\frac{a}{a-n}\)if\(0\ltn\lta\) \(\E(Z^n)=\infty\)if\(n\gea\) Proof: Notethat \[E(Z^n)=\int_1^\inftyz^n\frac{a}{z^{a+1}}dz=\int_1^\inftyaz^{-(a+1-n)}dz\] Theintegraldivergesto\(\infty\)if\(a+1-n\le1\)andevaluatesto\(\frac{a}{a-n}\)if\(a+1-n\gt1\). Itfollowsthatthemomentgeneratingfunctionof\(Z\)cannotbefiniteonanyintervalabout0. Inparticular,themeanandvarianceof\(Z\)are \(\E(Z)=\frac{a}{a-1}\)if\(a\gt1\) \(\var(Z)=\frac{a}{(a-1)^2(a-2)}\)if\(a\gt2\) Proof: Thisresultsfollowfromthegeneralmomentformulaaboveandthecomputationalformula\(\var(Z)=\E\left(Z^2\right)-[E(Z)]^2\). Inthespecialdistributionsimulator,selecttheParetodistribution.Varytheparametersandnotetheshapeandlocationofthemean\(\pm\)standarddeviationbar.Foreachofthefollowingparametervalues,runthesimulation1000timesandnotethebehavioroftheempiricalmoments: \(a=1\) \(a=2\) \(a=3\) Theskewnessandkurtosisof\(Z\)areasfollows: If\(a\gt3\), \[\skw(Z)=\frac{2(1+a)}{a-3}\sqrt{1-\frac{2}{a}}\] If\(a\gt4\), \[\kur(Z)=\frac{3(a-2)(3a^2+a+2)}{a(a-3)(a-4)}\] Proof: Theseresultsfollowfromthestandardcomputationalformulasforskewnessandkurtosis,andthefirst4momentsof\(Z\)givenabove. Sothedistributionispositivelyskewedand\(\skw(Z)\to2\)as\(a\to\infty\)while\(\skw(Z)\to\infty\)as\(a\downarrow3\).Similarly,\(\kur(Z)\to9\)as\(a\to\infty\)and\(\kur(Z)\to\infty\)as\(a\downarrow4\).Recallthattheexcesskurtosisof\(Z\)is \[\kur(Z)-3=\frac{3(a-2)(3a^2+a+2)}{a(a-3)(a-4)}-3=\frac{6(a^3+a^2-6a-1)}{a(a-3)(a-4)}\] RelatedDistributions ThebasicParetodistributionisinvariantunderpositivepowersoftheunderlyingvariable. Supposethat\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\in(0,\infty)\)andthat\(n\in(0,\infty)\).Then\(W=Z^n\)hasthebasicParetodistributionwithshapeparameter\(a/n\). Proof: WeusetheCDFof\(Z\)givenabove. \[\P(W\lew)=\P\left(Z\lew^{1/n}\right)=1-\frac{1}{w^{a/n}},\quadw\in[1,\infty)\] Asafunctionof\(w\),thisistheParetoCDFwithshapeparameter\(a/n\). Inparticular,if\(Z\)hasthestandardParetodistributionand\(a\in(0,\infty)\),then\(Z^{1/a}\)hasthebasicParetodistributionwithshapeparameter\(a\).Thus,allbasicParetovariablescanbeconstructedfromthestandardone. ThebasicParetodistributionhasareciprocalrelationshipwiththebetadistribution. Supposethat\(a\in(0,\infty)\). If\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\)then\(V=1/Z\)hasthebetadistributionwithleftparameter\(a\)andrightparameter1. If\(V\)hasthebetadistributionwithleftparameter\(a\)andrightparameter1,then\(Z=1/V\)hasthebasicParetodistributionwithshapeparameter\(a\). Proof: Wewillusethestandardchangeofvariablestheorem.Thetransformationsare\(v=1/z\)and\(z=1/v\)for\(z\in[1,\infty)\)and\(v\in(0,1]\).Theseareinversesofeachanother.Let\(g\)and\(h\)denotePDFsof\(Z\)and\(V\)respectively. Westartwith\(g(z)=a\big/z^{a+1}\)for\(z\in[1,\infty)\),thePDFof\(Z\)givenabove.Then \[h(v)=g(z)\left|\frac{dz}{dv}\right|=\frac{a}{(1/v)^{a+1}}\frac{1}{v^2}=av^{a-1},\quadv\in(0,1]\] whichisthePDFofthebetadistributionwithleftparameter\(a\)andrightparameter1. Westartwith\(h(v)=av^{a-1}\)for\(v\in(0,1]\).Then \[g(z)=h(v)\left|\frac{dv}{dz}\right]=a\left(\frac{1}{z}\right)^{a-1}\frac{1}{z^2}=\frac{a}{z^{a+1}},\quadz\in[1,\infty)\] whichisthePDFofthebasicParetodistributionwithshapeparameter\(a\). ThebasicParetodistributionhastheusualconnectionswiththestandarduniformdistributionbymeansofthedistributionfunctionandquantilefunctioncomputedabove. Supposethat\(a\in(0,\infty)\). If\(U\)hasthestandarduniformdistributionthen\(Z=1\big/U^{1/a}\)hasthebasicParetodistributionwithshapeparameter\(a\). If\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\)then\(U=1\big/Z^a\)hasthestandarduniformdistribution. Proof: If\(U\)hasthestandarduniformdistribution,thensodoes\(1-U\).Hence\(Z=G^{-1}(1-U)=1\big/U^{1/a}\)hasthebasicParetodistributionwithshapeparameter\(a\). If\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\),then\(G(Z)\)hasthestandarduniformdistribution.Butthen\(U=1-G(Z)=1\big/Z^a\)alsohasthestandarduniformdistribution. Sincethequantilefunctionhasasimpleclosedform,thebasicParetodistributioncanbesimulatedusingtherandomquantilemethod. OpentherandomquantileexperimentandselectedtheParetodistribution.Varytheshapeparameterandnotetheshapeofthedistributionandprobabilitydensityfunctions.Forselectedvaluesoftheparameter,runtheexperiment1000timesandcomparetheempiricaldensityfunction,mean,andstandarddeviationtotheirdistributionalcounterparts. ThebasicParetodistributionalsohassimpleconnectionstotheexponentialdistribution. Supposethat\(a\in(0,\infty)\). If\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\),then\(T=\lnZ\)hastheexponentialdistributionwithrateparameter\(a\). If\(T\)hastheexponentialdistributionwithrateparameter\(a\),then\(Z=e^T\)hasthebasicParetodistributionwithshapeparameter\(a\). Proof: WeusetheParetoCDFgivenaboveandtheCDFoftheexponentialdistribution. If\(t\in[0,\infty)\)then \[\P(T\let)=\P\left(Z\lee^t\right)=1-\frac{1}{\left(e^t\right)^a}=1-e^{-at}\] whichistheCDFoftheexponentialdistributionwithrateparameter\(a\). If\(z\in[1,\infty)\)then \[\P(Z\lez)=\P(T\le\lnz)=1-\exp(-a\lnz)=1-\frac{1}{z^z}\] whichistheCDFofthebasicParetodistributionwithshapeparameter\(a\). TheGeneralParetoDistribution Aswithmanyotherdistributionsthatgovernpositivevariables,theParetodistributionisoftengeneralizedbyaddingascaleparameter.Recallthatascaletransformationoftencorrespondstoachangeofunits(dollarsintoEuros,forexample)andthussuchtransformationsareofbasicimportance. Supposethat\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\in(0,\infty)\)andthat\(b\in(0,\infty)\).Randomvariable\(X=bZ\)hastheParetodistributionwithshapeparameter\(a\)andscaleparameter\(b\). Notethat\(X\)hasacontinuousdistributionontheinterval\([b,\infty)\). DistributionFunctions Supposeagainthat\(X\)hastheParetodistributionwithshapeparameter\(a\in(0,\infty)\)andscaleparameter\(b\in(0,\infty)\). \(X\)hasdistributionfunction\(F\)givenby \[F(x)=1-\left(\frac{b}{x}\right)^a,\quadx\in[b,\infty)\] Proof: Recallthat\(F(x)=G\left(\frac{x}{b}\right)\)for\(x\in[b,\infty)\)where\(G\)istheCDFofthebasicdistributionwithshapeparameter\(a\). \(X\)hasprobabilitydensityfunction\(f\)givenby \[f(x)=\frac{ab^a}{x^{a+1}},\quadx\in[b,\infty)\] Proof: Recallthat\(f(x)=\frac{1}{b}g\left(\frac{x}{b}\right)\)for\(x\in[b,\infty)\)where\(g\)isthePDFofthebasicdistributionwithshapeparameter\(a\). OpenthespecialdistributionsimulatorandselecttheParetodistribution.Varytheparametersandnotetheshapeandlocationoftheprobabilitydensityfunction.Forselectedvaluesoftheparameters,runthesimulation1000timesandcomparetheempiricaldensityfunctiontotheprobabilitydensityfunction. \(X\)hasquantilefunction\(F^{-1}\)givenby \[F^{-1}(p)=\frac{b}{(1-p)^{1/a}},\quadp\in[0,1)\] Thefirstquartileis\(q_1=b\left(\frac{4}{3}\right)^{1/a}\). Themedianis\(q_2=b2^{1/a}\). Thethirdquartileis\(q_3=b4^{1/a}\). Proof: Recallthat\(F^{-1}(p)=bG^{-1}(p)\)for\(p\in[0,1)\)where\(G^{-1}\)isthequantilefunctionofthebasicdistributionwithshapeparameter\(a\). OpenthespecialdistributioncalculatorandselecttheParetodistribution.Varytheparametersandnotetheshapeandlocationoftheprobabilitydensityanddistributionfunctions.Forselectedvaluesoftheparameters,computeafewvaluesofthedistributionandquantilefunctions. Moments Supposeagainthat\(X\)hastheParetodistributionwithshapeparameter\(a\in(0,\infty)\)andscaleparameter\(b\in(0,\infty)\) Themomentsof\(X\)aregivenby \(\E(X^n)=b^n\frac{a}{a-n}\)if\(0\ltn\lta\) \(\E(X^n)=\infty\)if\(n\gea\) Proof: Bydefinitionwecantake\(X=bZ\)where\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\).Bythelinearityofexpectedvalue,\(\E(X^n)=b^n\E(Z^n)\),sotheresultfollowsfromthemomentsof\(Z\)givenabove. Themeanandvarianceof\(X\)are \(\E(X)=b\frac{a}{a-1}\)if\(a\gt1\) \(\var(X)=b^2\frac{a}{(a-1)^2(a-2)}\)if\(a\gt2\) OpenthespecialdistributionsimulatorandselecttheParetodistribution.Varytheparametersandnotetheshapeandlocationofthemean\(\pm\)standarddeviationbar.Forselectedvaluesoftheparameters,runthesimulation1000timesandcomparetheempiricalmeanandstandarddeviationtothedistributionmeanandstandarddeviation. Theskewnessandkurtosisof\(X\)areasfollows: If\(a\gt3\), \[\skw(X)=\frac{2(1+a)}{a-3}\sqrt{1-\frac{2}{a}}\] If\(a\gt4\), \[\kur(X)=\frac{3(a-2)(3a^2+a+2)}{a(a-3)(a-4)}\] Proof: Recallthatskewnessandkurtosisaredefinedintermsofthestandardscore,andhenceareinvariantunderscaletransformations.Thustheskewnessandkurtosisof\(X\)arethesameastheskewnessandkurtosisof\(Z=X/b\)givenabove. RelatedDistributions SincetheParetodistributionisascalefamilyforfixedvaluesoftheshapeparameter,itistriviallyclosedunderscaletransformations. Supposethat\(X\)hastheParetodistributionwithshapeparameter\(a\in(0,\infty)\)andscaleparameter\(b\in(0,\infty)\).If\(c\in(0,\infty)\)then\(Y=cX\)hastheParetodistributionwithshapeparameter\(a\)andscaleparameter\(bc\). Proof: Bydefinitionwecantake\(X=bZ\)where\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\).Butthen\(Y=cX=(bc)Z\). TheParetodistributionisclosedunderpositivepowersoftheunderlyingvariable. Supposethat\(X\)hastheParetodistributionwithshapeparameter\(a\in(0,\infty)\)andscaleparameter\(b\in(0,\infty)\).If\(n\in(0,\infty)\)then\(Y=X^n\)hastheParetodistributionwithshapeparameter\(a/n\)andscaleparameter\(b^n\). Proof: Againwecanwrite\(X=bZ\)where\(Z\)hasthebasicParetodistributionwithshapeparameter\(a\).Thenfromthepowerresultabove\(Z^n\)hasthebasicParetodistibutionwithshapeparameter\(a/n\)andhence\(Y=X^n=b^nZ^n\)hastheParetodistributionwithshapeparameter\(a/n\)andscaleparameter\(b^n\). AllParetovariablescanbeconstructedfromthestandardone.If\(Z\)hasthestandardParetodistributionand\(a,\,b\in(0,\infty)\)then\(X=bZ^{1/a}\)hastheParetodistributionwithshapeparameter\(a\)andscaleparameter\(b\). Asbefore,theParetodistributionhastheusualconnectionswiththestandarduniformdistributionbymeansofthedistributionfunctionandquantilefunctiongivenabove. Supposethat\(a,\,b\in(0,\infty)\). If\(U\)hasthestandarduniformdistributionthen\(X=b\big/U^{1/a}\)hastheParetodistributionwithshapeparameter\(a\)andscaleparameter\(b\). If\(X\)hastheParetodistributionwithshapeparameter\(a\)andscaleparameter\(b\),then\(U=(b/X)^a\)hasthestandarduniformdistribution. Proof: If\(U\)hasthestandarduniformdistribution,thensodoes\(1-U\).Hence\(X=F^{-1}(1-U)=b\big/U^{1/a}\)hastheParetodistributionwithshapeparameter\(a\)andscaleparameter\(b\). If\(X\)hastheParetodistributionwithshapeparameter\(a\)andscaleparameter\(b\),then\(F(X)\)hasthestandarduniformdistribution.Butthen\(U=1-F(X)=(b/X)^a\)alsohasthestandarduniformdistribution. Again,sincethequantilefunctionhasasimpleclosedform,thebasicParetodistributioncanbesimulatedusingtherandomquantilemethod. OpentherandomquantileexperimentandselectedtheParetodistribution.Varytheparametersandnotetheshapeofthedistributionandprobabilitydensityfunctions.Forselectedvaluesoftheparameters,runtheexperiment1000timesandcomparetheempiricaldensityfunction,mean,andstandarddeviationtotheirdistributionalcounterparts. TheParetodistributionisclosedwithrespecttoconditioningonaright-tailevent. Supposethat\(X\)hastheParetodistributionwithshapeparameter\(a\in(0,\infty)\)andscaleparameter\(b\in(0,\infty)\).For\(c\in[b,\infty)\),theconditionaldistributionof\(X\)given\(X\gec\)isParetowithshapeparameter\(a\)andscaleparameter\(c\). Proof: Notsurprisingly,itsbesttouseright-taildistributionfunctions.Recallthatthisisthefunction\(F^c=1-F\)where\(F\)istheordinaryCDFgivenabove.If\(x\gec\),them \[\P(X\gtx\midX\gtc)=\frac{\P(X\gtx)}{\P(X\gtc)}=\frac{(b/x)^a}{(b/c)^a}=(c/x)^a\] Finally,theParetodistributionisageneralexponentialdistributionwithrespecttotheshapeparameter,forafixedvalueofthescaleparameter. Supposethat\(X\)hastheParetodistributionwithshapeparameter\(a\in(0,\infty)\)andscaleparameter\(b\in(0,\infty)\).Forfixed\(b\),thedistributionof\(X\)isageneralexponentialdistributionwithnaturalparameter\(-(a+1)\)andnaturalstatistic\(\lnX\). Proof: Thisfollowsfromthedefinitionofthegeneralexponentialfamily,sincethepdfabovecanbewrittenintheform \[f(x)=ab^a\exp[-(a+1)\lnx],\quadx\in[b,\infty)\] ComputationalExercises SupposethattheincomeofacertainpopulationhastheParetodistributionwithshapeparameter3andscaleparameter1000.Findeachofthefollowing: Theproportionofthepopulationwithincomesbetween2000and 4000. Themedianincome. Thefirstandthirdquartilesandtheinterquartilerange. Themeanincome. Thestandarddeviationofincome. The90thpercentile. Answer: \(\P(2000\ltX\lt4000)=0.1637\)sotheproportionis16.37% \(Q_2=1259.92\) \(Q_1=1100.64\),\(Q_3=1587.40\),\(Q_3-Q_1=486.76\) \(\E(X)=1500\) \(\sd(X)=866.03\) \(F^{-1}(0.9)=2154.43\)