1 Kaplan-Meier estimators of interpoint distance d

1Kaplan-MeierestimatorsofinterpointdistancedistributionsforspatialpointprocessesAdrianBaddeleyCentreforMathematicsandComputerScienceKruislaan413,1098SJAmsterdamTheNetherlandsDepartmentofMathematicsandComputerScienceUniversityofLeidenTheNetherlandsRichardD.GillMathematicalInstitute,UniversityofUtrecht,Budapestlaan6,3584CDUtrechtTheNetherlandsWhenaspatialpointprocessisobservedthroughaboundedwindow,edgeeectshampertheestimationofcharacteristicssuchastheemptyspacefunctionF,thenearestneighbourdistancedistributionG,andthesecondordermomentfunctionK.Hereweproposeandstudyproduct-limittypeestimatorsofF;GandKbasedontheanalogywithcensoredsurvivaldata:thedistancefromaxedpointtothenearestpointoftheprocessisright-censoredbyitsdistancetotheboundaryofthewindow.Theresultingestimatorshavearatio-unbiasednesspropertythatisstandardinspatialstatistics.Theestimatorsarestronglyconsistentwhenthereareindependentreplicationsorwhenthesamplingwindowbecomeslarge.Insimulationstheseestimatorsaregenerallymoreecientthanexistingestimators.WegivesomeasymptotictheoryforlargesamplingwindowsandforsparsePoissonprocesses.Varianceestimatorsareproposed.AMSMathematicsSubjectClassication(1991Revision):primary:62G05;secondary:62H11,60D05.KeyWordsandPhrases:bordercorrectionmethod,dilation,distancetransform,edgecorrec-tions,edgeeects,emptyspacestatistic,erosion,functionaldelta-method,inuencefunction,K-function,localknowledgeprinciple,nearest-neighbourdistancedistribution,productin-tegration,reducedsampleestimator,sparseintensityasymptotics,spatialstatistics,survivaldata.1IntroductionTheexploratorydataanalysisofobservationsofaspatialpointprocessoftenstartswiththeestimationofcertaindistancedistributions:F,thedistributionofthedistancefromanarbitrarypointinspacetothenearestpointoftheprocess;G,thedistributionofthedistance2fromatypicalpointoftheprocesstothenearestotherpointoftheprocess;andK(t),theexpectednumberofotherpointswithindistancetofatypicalpointoftheprocess,dividedbytheintensity.EquivalentlyKisthesumoveralln=1;2;:::ofthedistributionofthedistancefromatypicalpointoftheprocesstothenthnearestpoint.PopularnamesforF;GandKaretheemptyspacefunction,thenearestneighbourdistancedistribution,andthesecondmomentfunction.ForahomogeneousPoissonprocessF;GandKtakeknownfunctionalforms,anddeviationsofestimatesofF;G;Kfromtheseformsaretakenasindicationsof‘clustered’or‘inhibited’alternatives[7,22,23].However,theestimationofF;GandKishamperedbyedgeeectsarisingbecausethepointprocessisobservedwithinaboundedwindowW.EssentiallythedistancefromagivenreferencepointtothenearestpointoftheprocessiscensoredbyitsdistancetotheboundaryofW.Edgeeectsbecomerapidlymoresevereasthedimensionofspaceincreases,orasthedistancetincreases.Traditionallyinspatialstatistics,oneusesedge-correctedestimatorswhichareweightedempiricaldistributionsoftheobserveddistances.Thesimplestapproachisthe\bordermethod[23]inwhichwerestrictattention(whenestimatingF;GorKatdistancet)tothosereferencepointslyingmorethantunitsawayfromtheboundaryofW.Thesearethepointsxforwhichdistancesuptotareobservedwithoutcensoring.Thisapproachissometimesalsojustiedbyappealingtothe\localknowledgeprincipleofmathematicalmorphology[28,pp.49,233].However,thebordermethodthrowsawayanappreciablenumberofpoints;inthreedimensions[2]itseemstobeunacceptablywasteful,especiallywhenestimatingG.Inmoresophisticatededgecorrections,theweightc(x;y)attachedtotheobserveddis-tancejjxyjjbetweentwopointsx;yisthereciprocaloftheprobabilitythatthisdistancewillbeobservedundercertaininvarianceassumptions(stationarityundertranslationand/orrotation).CorrectionsofthistypewererstsuggestedbyMiles[20]anddevelopedbyRipley,Lantuejoul,Hanisch,Ohserandothers[7,14,21,24,22],[23,chap.3],[28,p.246],[30,pp.122{131].NowtheestimationproblemforF;GandKwhenobservingapointprocessthroughaboundedwindowWhasaclearanalogy,alreadyimplicitlydrawnabove,totheestimationofasurvivalfunctionbasedonasampleofrandomlycensoredsurvivaltimes.Thispaperdevelopstheanalogy,andproposesKaplan-Meier[15]orproduct-limitestimatorsforF;GandK.Sincetheobserved,censoreddistancesarehighlyinterdependent,classicaltheoryfromsurvivalanalysishaslittletosayaboutstatisticalpropertiesofthenewestimators.Onemayhoweverhopethatthenewestimatorsarebetterthantheclassicaledgecorrections,asinthesurvivalanalysissituationtheKaplan-Meierestimatorhasvariouslarge-sampleoptimalityproperties.Infactthebordermethodforedgecorrection,describedabove,isanalogoustotheso-calledreducedsampleestimator,aninecientcompetitortotheKaplan-Meierestimatorobtainedusingonlythoseobservationsforwhichthecensoringtimeisatleasttwhenestimatingtheprobabilityofsurvivaltotimet.Surprisinglytheanalogybetweenedgeeectsforpointprocessesandcensoringofsurvivaltimesdoesnotseemtohavebeennotedbefore.Laslett[16,17]notedthatwhenaspatiallinesegmentprocessisobservedwithinaboundedwindow,theresultingedgeeectsontheobservedlinesegmentlengthscanbecomparedtocensoringofsurvivaltimes.Howeverin3thatcasetheanalogyisnotespeciallyhelpful:aKaplan-Meiertypeestimatorforthesegmentlengthdistributi