7Lecture7:OrderStatistics7.1MotivationLetX=(X1;X2;:::;Xn)denotearandomsamplefromapopulationwithacontinuousdistributionfunctionFX.SinceFXisassumedtobecontinuous,theprobabilityofanytwooftheserandomvariablesassumingthesamevalueiszero.AfterreorderingthenvalueswegetX(1)≤X(2)≤···≤X(n))inwhich,asmentioned,the≤signcouldalsobereplacedby<.ThesevaluesarecollectivelytermedtheorderstatisticoftherandomsampleX=(X1;X2;:::;Xn).ThesubjectoforderstatisticsgenerallydealswithpropertiesofX(r)(r=1;2;:::;n)whichiscalledther-thorderstatistic.Orderstatisticsareparticularlyusefulinnonparametricstatisticsbecauseofthefol-lowing:Theorem7.1.(Probability-integraltransformation).IftherandomvariableXhasacontinuouscdfFXthentherandomvariableY=FX(X)hastheuniformprobabilitydistributionovertheinterval(0,1).Further,givenasampleX=(X1;X2;:::;Xn)ofni.i.d.randomvariableswithcdfFX,thetransformationU(r)=FX(X(r))producesarandomvariableU(r)whichisther-thorderstatisticfromtheuniformpopulationin(0,1),regardlessofwhatFXis,i.e.U(r)isdistribution-free.Proof:ItisyourtextbookandwasalsodiscussedintheintroductoryLecture1.Note:Theabovetheoremhasalsoanextremelyimportantpracticalapplicationinthegeneration(computersimulation)ofobservationsfromanyspecificcontinuousdistribu-tionfunction.Thereareseveralwell-developeduniformrandomnumbergeneratorsthatimplementmethodstogeneratesequencesofuniformin(0,1)pseudo-randomnum-bers.Thesenumbersarepseudosinceinfacttheyaregeneratedbyadeterministicalgo-rithm(thereforearenotrandom)butlookasrandom(hencethewordpseudo-random)inthesensethattheypassusualstatisticaltestsaboutrandomnessofthegeneratedse-quence.Everyprogramsystem(Fortran,SPLUS,C,SAS,etc.)hassuchuniformrandomnumbergeneratorsandwewillnotdiscusstheirspecificimplementationhere.WhatwewouldliketodiscussishowwecouldusetheseuniformrandomnumbergeneratorstogeneraterandomnumberswitharbitrarycontinuouscumulativedistributionfunctionFX:Theansweris:1)GenerateYasuniformlydistributedin(0,1)usingtheuniformrandomnumbergener