Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto Q–Q plots), mean residual life plotsBeirlant, J., Teugels, J. L., Vynckier, P. (1996) Practical Analysis of Extreme Values, Leuven: Leuven University PressColes, S. (2001) An introduction to statistical modeling of extreme values. Springer-Verlag, London. and log–log plots. Another, more robust graphical method uses bundles of residual quantile functions. (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data".
althOUgh more sophistIcaTed aND ROBust mEtHodS havE been PrOPosEd, thE Most Frequently usEd GrapHical MetHoDs OF ideNtifyinG POwer-law PRobabiLity distrIbutIons using raNdom SAmpleS arE PArEto QuantiLe-qUANTiLe pLots (OR pareTo q–q PlOtS), mEAn reSidual liFe plotsBeirLanT, J., teugeLS, j. l., vyncKIeR, p. (1996) prACtIcAl ANALysIs of extrEme ValuEs, leuVen: leuvEN uNiverSitY presscolEs, s. (2001) aN INtroduCtIon to stATisticaL MODeling of eXTreMe vAlUes. sprinGer-verlag, lOndon. and Log–lOg ploTS. AnotHER, MoRe robust GrapHiCAL Method uSeS bundleS of ResiDUaL qUanTILe fUnctIOnS. (pLeaSE keep iN MiNd that PowEr-LaW distrIbUTiOns are Also called Pareto-typE distriButionS.) it iS AssUMed heRE that a ranDom saMpLe Is OBTAined from A ProBabIlIty diStRibUTion, anD thAt we wAnt to kNOw if tHe tail OF The distRiButIon FOLLowS a poWer lAW (In oTHer WordS, we waNT to knOw If thE distribuTiOn has a "pareto Tail"). here, thE rAndOm SaMple is CAllEd "THe DATa".