Coordination of spatially balanced samples
Section 2. Notation

Let U 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaaigdaaeqaaa aa@33B2@ and U 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaaikdaaeqaaa aa@33B3@ be a population (subject to change over time) at time 1 and time 2, respectively, or consider that U 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaaigdaaeqaaa aa@33B2@ and U 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaaikdaaeqaaa aa@33B3@ are two overlapping populations. Consider samples s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ and s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaa aa@33D1@ drawn from U 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaaigdaaeqaaa aa@33B2@ and U 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaaikdaaeqaaO GaaGilaaaa@3473@ using the sampling designs p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaa aa@33CD@ and p 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaSbaaSqaaiaaikdaaeqaaO GaaGilaaaa@348E@ respectively. No restriction about the sampling designs p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaa aa@33CD@ and p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaSbaaSqaaiaaikdaaeqaaa aa@33CE@ is necessary to introduce the definitions in this section: they can be fixed or random size sampling designs, with or without replacement.

Let U = U 1 U 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbGaaGypaiaadwfadaWgaaWcba GaaGymaaqabaGccaaMe8UaeyOkIGSaaGjbVlaadwfadaWgaaWcbaGa aGOmaaqabaGccaaIUaaaaa@3C9B@ We call U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbaaaa@32CB@ the “overall population”. The set of labels of the units in U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbaaaa@32CB@ is { 1, 2, , i , , N } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaGadaqaaiaaigdacaaISaGaaGjbVl aaikdacaaISaGaaGjbVlablAciljaaiYcacaaMe8UaamyAaiaaiYca caaMe8UaeSOjGSKaaGilaiaaysW7caWGobaacaGL7bGaayzFaaGaaG Olaaaa@45A5@ We define on U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbaaaa@32CB@ the joint sampling design p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbaaaa@32E6@ used to select a couple ( s 1 , s 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadohadaWgaaWcbaGaaG ymaaqabaGccaaISaGaaGjbVlaadohadaWgaaWcbaGaaGOmaaqabaaa kiaawIcacaGLPaaacaaIUaaaaa@3A48@ The samples s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ and s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaa aa@33D1@ are coordinated if p ( s 1 , s 2 ) p 1 ( s 1 ) p 2 ( s 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbGaaGPaVpaabmaabaGaam4Cam aaBaaaleaacaaIXaaabeaakiaaiYcacaaMe8Uaam4CamaaBaaaleaa caaIYaaabeaaaOGaayjkaiaawMcaaiabgcMi5kaadchadaWgaaWcba GaaGymaaqabaGcdaqadaqaaiaadohadaWgaaWcbaGaaGymaaqabaaa kiaawIcacaGLPaaacaWGWbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaae aacaWGZbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaaiil aaaa@4939@ that is the samples are not drawn independently (see Cotton and Hesse, 1992; Mach, Reiss and Şchiopu-Kratina, 2006). Let π i 1 = P ( i s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaig daaeqaaOGaaGypaiaadcfadaqadaqaaiaadMgacqGHiiIZcaWGZbWa aSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@3D0D@ and π i 2 = P ( i s 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaik daaeqaaOGaaGypaiaadcfadaqadaqaaiaadMgacqGHiiIZcaWGZbWa aSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@3D0F@ be the first-order inclusion probabilities of unit i U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyicI4Saamyvaaaa@353D@ in the first and second sample, respectively. It follows that π i 1 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaig daaeqaaOGaaGypaiaaicdaaaa@370E@ if i U 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyycI8SaamyvamaaBaaale aacaaIXaaabeaaaaa@3626@ and π i 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaik daaeqaaOGaaGypaiaaicdaaaa@370F@ if i U 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGjbVlaayIW7cqGHjiYZca aMe8UaaGjcVlaadwfadaWgaaWcbaGaaGOmaaqabaGccaaIUaaaaa@3D25@ Thus, it is not necessary to identify explicitly the subpopulation memberships.

Let π i , 12 = P ( i s 1 , i s 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaiY cacaaMe8UaaGymaiaaikdaaeqaaOGaaGypaiaadcfadaqadaqaaiaa dMgacqGHiiIZcaWGZbWaaSbaaSqaaiaaigdaaeqaaOGaaGzaVlaaiY cacaaMe8UaamyAaiabgIGiolaadohadaWgaaWcbaGaaGOmaaqabaaa kiaawIcacaGLPaaaaaa@4835@ be the joint inclusion probability of unit i U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyicI4Saamyvaaaa@353D@ in both samples s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ and s 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaO GaaGOlaaaa@3493@ If the samples s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ and s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaa aa@33D1@ are selected independently, π i , 12 = π i 1 π i 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaiY cacaaMe8UaaGymaiaaikdaaeqaaOGaaGypaiabec8aWnaaBaaaleaa caWGPbGaaGymaaqabaGccqaHapaCdaWgaaWcbaGaamyAaiaaikdaae qaaOGaaGzaVlaaiYcaaaa@42CC@ for all i U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyicI4Saamyvaiaai6caaa a@35F5@

Let c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGJbaaaa@32D9@ be the overlap between s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ and s 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaO GaaGzaVlaaiYcaaaa@361B@ which represents the number of common units of the two samples; it is in most of the cases a random variable. The coordination degree of s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ and s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaa aa@33D1@ is measured by the expected overlap

E ( c ) = i U π i , 12 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGfbWaaeWaaeaacaWGJbaacaGLOa GaayzkaaGaaGjbVlaaysW7caaI9aGaaGjbVlaaysW7daaeqbqaaiab ec8aWnaaBaaaleaacaWGPbGaaGilaiaaysW7caaIXaGaaGOmaaqaba aabaGaamyAaiabgIGiolaadwfaaeqaniabggHiLdGccaaISaaaaa@48DB@

where π i , 12 = P ( i s 1 , i s 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaiY cacaaMe8UaaGymaiaaikdaaeqaaOGaaGypaiaadcfadaqadaqaaiaa dMgacqGHiiIZcaWGZbWaaSbaaSqaaiaaigdaaeqaaOGaaGzaVlaaiY cacaaMe8UaamyAaiabgIGiolaadohadaWgaaWcbaGaaGOmaaqabaaa kiaawIcacaGLPaaacaaIUaaaaa@48ED@ By using the Fréchet bounds of the joint probability π i , 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaaiY cacaaMe8UaaGymaiaaikdaaeqaaaaa@3882@ it follows that

i U max ( 0, π i 1 + π i 2 1 ) E ( c ) = i U π i , 12 i U min ( π i 1 , π i 2 ) . ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaaeqbqaaiGac2gacaGGHbGaaiiEam aabmaabaGaaGimaiaaiYcacaaMe8UaeqiWda3aaSbaaSqaaiaadMga caaIXaaabeaakiabgUcaRiabec8aWnaaBaaaleaacaWGPbGaaGOmaa qabaGccqGHsislcaaIXaaacaGLOaGaayzkaaaaleaacaWGPbGaeyic I4Saamyvaaqab0GaeyyeIuoakiabgsMiJkaadweadaqadaqaaiaado gaaiaawIcacaGLPaaacaaMe8UaaGjbVlaai2dacaaMe8UaaGjbVpaa qafabaGaeqiWda3aaSbaaSqaaiaadMgacaaISaGaaGjbVlaaigdaca aIYaaabeaaaeaacaWGPbGaeyicI4Saamyvaaqab0GaeyyeIuoakiab gsMiJoaaqafabaGaciyBaiaacMgacaGGUbWaaeWaaeaacqaHapaCda WgaaWcbaGaamyAaiaaigdaaeqaaOGaaGzaVlaaiYcacaaMe8UaeqiW da3aaSbaaSqaaiaadMgacaaIYaaabeaaaOGaayjkaiaawMcaaaWcba GaamyAaiabgIGiolaadwfaaeqaniabggHiLdGccaaIUaGaaGzbVlaa ywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaigdacaGGPaaaaa@8171@

In negative coordination one wants to achieve the left bound in expression (2.1), that is i U max ( 0, π i 1 + π i 2 1 ) = E ( c ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaaeqaqaaiGac2gacaGGHbGaaiiEam aabmaabaGaaGimaiaaiYcacaaMe8UaeqiWda3aaSbaaSqaaiaadMga caaIXaaabeaakiabgUcaRiabec8aWnaaBaaaleaacaWGPbGaaGOmaa qabaGccqGHsislcaaIXaaacaGLOaGaayzkaaaaleaacaWGPbGaeyic I4Saamyvaaqab0GaeyyeIuoakiaai2dacaWGfbWaaeWaaeaacaWGJb aacaGLOaGaayzkaaGaaGilaaaa@4CFF@ while in positive coordination the right bound, that is E ( c ) = i U min ( π i 1 , π i 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGfbWaaeWaaeaacaWGJbaacaGLOa GaayzkaaGaaGypamaaqababaGaciyBaiaacMgacaGGUbWaaeWaaeaa cqaHapaCdaWgaaWcbaGaamyAaiaaigdaaeqaaOGaaGzaVlaaiYcaca aMe8UaeqiWda3aaSbaaSqaaiaadMgacaaIYaaabeaaaOGaayjkaiaa wMcaaaWcbaGaamyAaiabgIGiolaadwfaaeqaniabggHiLdGccaaIUa aaaa@4B45@ Thus, to optimize the sample coordination process, the goal is to achieve these bounds, prior to coordination type, positive or negative. Using the terminology of Matei and Tillé (2005) the left side-part in (2.1) is called the Absolute Lower Bound (ALB) and the right side-part in (2.1) the Absolute Upper Bound (AUB).

The focus here is on sample coordination using PRNs. The PRN method was originally introduced by Brewer et al. (1972) to coordinate Poisson samples. Poisson sampling with PRNs reaches the Fréchet bounds given in equation (2.1). Yet, it results in a random sample size and does not provide spatially balanced samples. In order to achieve spatial balance, the local pivotal method (Grafström et al., 2012) and the spatially correlated Poisson sampling (Grafström, 2012) are used. Both sampling designs provide a good degree of spatial balance (see Grafström et al., 2012, for some empirical results). Moreover, since both are fixed size π ps MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCcaqGWbGaae4Caaaa@3597@ sampling designs (probability proportional to size sampling, see Särndal, Swensson and Wretman, 1992, page 90), the precision of the estimators is in general improved compared to Poisson sampling.

In what follows, we consider the sampling designs p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaa aa@33CD@ and p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaSbaaSqaaiaaikdaaeqaaa aa@33CE@ to be without replacement, and the expected sample sizes of s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ and s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaa aa@33D1@ are denoted by n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaaigdaaeqaaa aa@33CB@ and n 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaaikdaaeqaaO GaaGzaVlaaiYcaaaa@3616@ respectively.


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