Using balanced sampling in creel surveys
Section 4. Comparison of the cube method and the rejective algorithm

Chauvet et al. (2015) have studied the cube method and the rejective algorithm by examining different aspects of these balancing techniques. They balanced on continuous auxiliary variables and they documented how the balancing algorithm impacted the selection probabilities and the sampling properties of estimators of population totals. The goal of this section is to compare the two sampling algorithms in a resource inventory where the balancing equations only involve indicator variables. This comparison is carried out in the context of a simplified creel survey with a stratified two stage design. The days represent strata h = 1, , H , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGObGaaGypaiaaigdacaaISaGaeS OjGSKaaGilaiaadIeacaGGSaaaaa@37FF@ the sectors are defined as primary units i = 1,2,3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdacaaISaGaaG OmaiaaiYcacaaIZaaaaa@36DA@ and sites, indexed by j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaiilaaaa@3324@ are the secondary units. This sampling plan is similar to the design exposed in Section 3.1 except that periods and subperiods do not enter in the sampling design.

On each day two out of 3 sectors are selected and within each one 2 sites are sampled; thus 4 units are selected each day. The site importance variable x i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@348B@ determines the inclusion probabilities π h i j = ( 2 x i / x ) × ( 2 x i j / x i ) = π h i × π h j | i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamiAaiaadM gacaWGQbaabeaakiaai2dadaqadaqaamaalyaabaGaaGOmaiaadIha daWgaaWcbaGaamyAaiabgkci3cqabaaakeaacaWG4bWaaSbaaSqaai abgkci3kabgkci3cqabaaaaaGccaGLOaGaayzkaaGaey41aq7aaeWa aeaadaWcgaqaaiaaikdacaWG4bWaaSbaaSqaaiaadMgacaWGQbaabe aaaOqaaiaadIhadaWgaaWcbaGaamyAaiabgkci3cqabaaaaaGccaGL OaGaayzkaaGaaGypaiabec8aWnaaBaaaleaacaWGObGaamyAaaqaba GccqGHxdaTcqaHapaCdaWgaaWcbaWaaqGaaeaacaWGObGaamOAaiaa yIW7aiaawIa7aiaayIW7caWGPbaabeaaaaa@5C86@ for the two stages. As two out of three units are selected at each level, the joint selection probabilities are completely determined by { ( π h i , π h j | i ) : i , j = 1,2,3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaGadaqaamaabmaabaGaeqiWda3aaS baaSqaaiaadIgacaWGPbaabeaakiaaiYcacqaHapaCdaWgaaWcbaWa aqGaaeaacaWGObGaamOAaiaayIW7aiaawIa7aiaayIW7caWGPbaabe aaaOGaayjkaiaawMcaaiaaiQdacaWGPbGaaGilaiaadQgacaaI9aGa aGymaiaaiYcacaaIYaGaaGilaiaaiodaaiaawUhacaGL9baaaaa@4AF6@ for the two stages; see the Appendix. If Z h i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGAbWaaSbaaSqaaiaadIgacaWGPb GaamOAaaqabaaaaa@355A@ stands for the indicator variables taking the value 1 if site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ is sampled on day h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGObaaaa@3272@ and 0 otherwise then the entries of 9 × 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaI5aGaey41aqRaaGyoaaaa@3522@ variance covariance matrix for { Z h i j : i , j = 1,2,3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaGadaqaaiaadQfadaWgaaWcbaGaam iAaiaadMgacaWGQbaabeaakiaayIW7caaI6aGaamyAaiaaiYcacaWG QbGaaGypaiaaigdacaaISaGaaGOmaiaaiYcacaaIZaaacaGL7bGaay zFaaaaaa@40E4@ are given by

Cov ( Z h i j , Z h i j ) = { π h i j π h i j 2 if i = i and j = j π h i π h j j | i π h i j π h i j if i = i and j j π h i i π h j | i π h j | i π h i j π h i j if i i ( 4.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGdbGaae4BaiaabAhadaqadaqaai aadQfadaWgaaWcbaGaamiAaiaadMgacaWGQbaabeaakiaaiYcacaWG AbWaaSbaaSqaaiaadIgaceWGPbGbauaaceWGQbGbauaaaeqaaaGcca GLOaGaayzkaaGaaGypamaaceaabaqbaeaabmGaaaqaaiabec8aWnaa BaaaleaacaWGObGaamyAaiaadQgaaeqaaOGaeyOeI0IaeqiWda3aa0 baaSqaaiaadIgacaWGPbGaamOAaaqaaiaaikdaaaaakeaacaqGPbGa aeOzaiaaysW7caWGPbGaaGypaiaadMgadaahaaWcbeqaaKqzGfGama i2gkdiIcaakiaaysW7caqGHbGaaeOBaiaabsgacaaMe8UaamOAaiaa i2dacaWGQbWaaWbaaSqabeaajugybiadaITHYaIOaaaakeaacqaHap aCdaWgaaWcbaGaamiAaiaadMgaaeqaaOGaaGPaVlabec8aWnaaBaaa leaacaWGObGaamOAaiqadQgagaqbaiaayIW7caGG8bGaaGjcVlaadM gaaeqaaOGaeyOeI0IaeqiWda3aaSbaaSqaaiaadIgacaWGPbGaamOA aaqabaGccaaMc8UaeqiWda3aaSbaaSqaaiaadIgacaWGPbGabmOAay aafaaabeaaaOqaaiaabMgacaqGMbGaaGjbVlaadMgacaaI9aGaamyA amaaCaaaleqabaqcLbwacWaGyBOmGikaaOGaaGjbVlaabggacaqGUb GaaeizaiaaysW7caWGQbGaeyiyIKRaamOAamaaCaaaleqabaqcLbwa cWaGyBOmGikaaaGcbaGaeqiWda3aaSbaaSqaaiaadIgacaWGPbGabm yAayaafaaabeaakiaaykW7cqaHapaCdaWgaaWcbaWaaqGaaeaacaWG ObGaamOAaiaayIW7aiaawIa7aiaayIW7caWGPbaabeaakiaaykW7cq aHapaCdaWgaaWcbaGaamiAaiqadQgagaqbaiaayIW7caGG8bGaaGjc VlqadMgagaqbaaqabaGccqGHsislcqaHapaCdaWgaaWcbaGaamiAai aadMgacaWGQbaabeaakiaaykW7cqaHapaCdaWgaaWcbaGaamiAaiaa dMgadaahaaadbeqaaiadaITHYaIOaaWccaWGQbWaaWbaaWqabeaacW aGyBOmGikaaaWcbeaaaOqaaiaabMgacaqGMbGaaGjbVlaadMgacqGH GjsUcaWGPbWaaWbaaSqabeaajugybiadaITHYaIOaaaaaaGccaGL7b aacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI0aGaaiOl aiaaigdacaGGPaaaaa@D6CD@

where π h i i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamiAaiaadM gacaWGPbWaaWbaaWqabeaacWaGyBOmGikaaaWcbeaaaaa@3950@ represents the joint selection probability of sectors i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@3273@ and i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbWaaWbaaSqabeaajugybiadaI THYaIOaaaaaa@364F@ on a single day, π h j | i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaWaaqGaaeaaca WGObGaamOAaiaayIW7aiaawIa7aiaayIW7caWGPbaabeaaaaa@3AF0@ is the probability for selecting site j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaiilaaaa@3324@ in sector i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaiilaaaa@3323@ at stage 2 and π h j j | i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamiAaiaadQ gaceWGQbGbauaacaaMi8UaaiiFaiaayIW7caWGPbaabeaaaaa@3B55@ is the joint selection probability of sites j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbaaaa@3274@ and j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbWaaWbaaSqabeaajugybiadaI THYaIOaaaaaa@3650@ in sector i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaiOlaaaa@3325@ All these probabilities are evaluated using the size measure x . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bGaaiOlaaaa@3334@ Details are available in the appendix, see also Ousmane Ida (2016). The corresponding matrix Var ( n ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGwbGaaeyyaiaabkhadaqadaqaai qad6gagaacaaGaayjkaiaawMcaaaaa@36C2@ in (2.3) is singular as one of the 9 constraints is redundant; thus in (2.3) a generalized inverse of the covariance matrix was used and γ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHZoWzdaahaaWcbeqaaiaaikdaaa GccaaMb8Uaaiilaaaa@3659@ in (2.3), was set equal to 2.73 and 7.34, the 5 th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaI1aWaaWbaaSqabeaacaqG0bGaae iAaaaaaaa@3453@ and the 50 th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaI1aGaaGimamaaCaaaleqabaGaae iDaiaabIgaaaaaaa@350D@ percentiles of the χ 8 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHhpWydaqhaaWcbaGaaGioaaqaai aaikdaaaaaaa@34E7@ distribution.

4.1  Simulations on the comparison of the cube method and of the rejective algorithm

To investigate the impact of the algorithm on the sampling properties of survey estimators we simulated, for each unit, a fishing effort for site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ on day h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGObGaaiilaaaa@3322@ y h i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadIgacaWGPb GaamOAaaqabaGccaGGSaaaaa@3633@ using independent Poisson random variables with mean 15 × x i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIXaGaaGynaiabgEna0kaadIhada WgaaWcbaGaamyAaiaadQgaaeqaaOGaaiOlaaaa@38D8@ The total fishing effort for site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ is then

Y U i j = h = 1 H y h i j . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGzbWaaSbaaSqaaiaadwfacaWGPb GaamOAaaqabaGccaaI9aWaaabCaeaacaWG5bWaaSbaaSqaaiaadIga caWGPbGaamOAaaqabaaabaGaamiAaiaai2dacaaIXaaabaGaamisaa qdcqGHris5aOGaaGOlaaaa@403F@

A calibrated estimator, as defined in Section 3.2, for the fishing effort in site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ is Y ^ i j = H y ¯ s i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamyAai aadQgaaeqaaOGaaGypaiaadIeacaaMi8UabmyEayaaraWaaSbaaSqa aiaadohacaWGPbGaamOAaaqabaGccaGGSaaaaa@3C7C@ the average fishing effort for the n i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@3481@ units sampled at site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ times H . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGibGaaiOlaaaa@3304@

To compare the balancing algorithms, we used designs with H = 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGibGaaGypaiaaigdacaaIYaaaaa@3490@ strata and two importance variables x , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bGaaiilaaaa@3332@ one with a small variation between site and one with a medium variation. Under each scenario we generated B = 100,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGcbGaaGypaiaabgdacaqGWaGaae imaiaabYcacaqGWaGaaeimaiaabcdaaaa@37F5@ random replications of a balanced sample by using the cube methods on one hand, and two rejective algorithms on the other. The inclusion probabilities for site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ was estimated by

π ^ i j = 1 B × H b = 1 B n i j ( b ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHapaCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaGccaaI9aWaaSaaaeaacaaIXaaabaGaamOqaiabgEna 0kaadIeaaaWaaabCaeaacaWGUbWaa0baaSqaaiaadMgacaWGQbaaba WaaeWaaeaacaWGIbaacaGLOaGaayzkaaaaaaqaaiaadkgacaaI9aGa aGymaaqaaiaadkeaa0GaeyyeIuoakiaai6caaaa@4637@

This estimator assumes that the inclusion probabilities π h i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamiAaiaadM gacaWGQbaabeaaaaa@3638@ are constant in h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGObGaaiOlaaaa@3324@ This holds true because the sample design is invariant to a relabelling of the days, see Section 3.1.

As argued in Section 3.2, the calibrated estimator Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamyAai aadQgaaeqaaaaa@347C@ is design unbiased under the two selection algorithms. We compare their standard deviations,

Sd Y ^ i j = { 1 B 1 b = 1 B ( Y ^ i j ( b ) Y ^ ¯ i j ) 2 } 1 / 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGtbGaaeizamaaBaaaleaaceWGzb GbaKaadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeaakiaai2dadaGa daqaamaalaaabaGaaGymaaqaaiaadkeacqGHsislcaaIXaaaamaaqa habaWaaeWaaeaaceWGzbGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaa daqadaqaaiaadkgaaiaawIcacaGLPaaaaaGccqGHsislceWGzbGbaK GbaebadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaWa aWbaaSqabeaacaaIYaaaaaqaaiaadkgacaaI9aGaaGymaaqaaiaadk eaa0GaeyyeIuoaaOGaay5Eaiaaw2haamaaCaaaleqabaWaaSGbaeaa caaIXaaabaGaaGOmaaaaaaGccaaMb8UaaGilaaaa@520A@

where Y ^ ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKGbaebadaWgaaWcbaGaam yAaiaadQgaaeqaaaaa@3493@ is the average of the B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGcbaaaa@324C@ simulated values. The sample size standard deviations were also calculated using (3.2). Observe that π ^ i j = n ¯ i j / H . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHapaCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaGccaaI9aWaaSGbaeaaceWGUbGbaebadaWgaaWcbaGa amyAaiaadQgaaeqaaaGcbaGaamisaaaacaGGUaaaaa@3ADF@ The simulation results are presented in Tables 4.1, 4.2 and 4.3.

Table 4.1
Comparison of the cube method (CM) and of two rejective algorithms (R 5% and R 50%) when x has a low variation
Table summary
This table displays the results of Comparison of the cube method (CM) and of two rejective algorithms (R 5% and R 50%) when x has a low variation (équation) (appearing as column headers).
CM R 5 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ R 50 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@
Sector Site x i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4bWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@36B8@ π i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacqaHapaCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3778@ π ^ ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaadaqdaaqaaiqbec8aWzaajaaaamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@3799@ Sd Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGtbGaaeizamaaBaaaleaaceWGzb GbaKaadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeaaaaa@389E@ π ^ ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaadaqdaaqaaiqbec8aWzaajaaaamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@3799@ Sd Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGtbGaaeizamaaBaaaleaaceWGzb GbaKaadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeaaaaa@389E@ π ^ ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaadaqdaaqaaiqbec8aWzaajaaaamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@3799@ Sd Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGtbGaaeizamaaBaaaleaaceWGzb GbaKaadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeaaaaa@389E@
i = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.500 0.500 16.56 0.503 16.86 0.505 17.40
j = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 22.20 0.329 23.35 0.328 25.07
j = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.500 0.500 23.99 0.503 24.47 0.505 25.15
i = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 25.80 0.329 26.93 0.326 29.11
j = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 33.97 0.329 35.54 0.326 38.28
j = 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 27.65 0.329 28.87 0.326 31.10
i = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.500 0.500 22.50 0.502 22.88 0.502 23.66
j = 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.500 0.500 20.02 0.502 20.20 0.502 20.94
j = 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 4 0.667 0.667 22.01 0.674 21.98 0.679 22.25
Table 4.2
Comparison of the cube method (CM) and of two rejective algorithms (R 5% and R 50%) when x has a medium variation
Table summary
This table displays the results of Comparison of the cube method (CM) and of two rejective algorithms (R 5% and R 50%) when x has a medium variation CM and (équation) (appearing as column headers).
CM R 5 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ R 50 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@
Sector Site x i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4bWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@36B8@ π i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacqaHapaCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3778@ π ^ ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaadaqdaaqaaiqbec8aWzaajaaaamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@3799@ Sd Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGtbGaaeizamaaBaaaleaaceWGzb GbaKaadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeaaaaa@389E@ π ^ ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaadaqdaaqaaiqbec8aWzaajaaaamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@3799@ Sd Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGtbGaaeizamaaBaaaleaaceWGzb GbaKaadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeaaaaa@389E@ π ^ ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaadaqdaaqaaiqbec8aWzaajaaaamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@3799@ Sd Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGtbGaaeizamaaBaaaleaaceWGzb GbaKaadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeaaaaa@389E@
i = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.500 0.500 25.52 0.505 25.78 0.507 26.60
j = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 25.25 0.330 26.26 0.329 28.16
j = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.500 0.500 21.12 0.505 21.36 0.507 22.03
i = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaikdaaaa@3619@ j = 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 1 0.167 0.167 29.17 0.158 32.45 0.149 31.19
j = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 13.73 0.329 14.38 0.326 15.49
j = 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 32.82 0.329 34.22 0.326 36.91
i = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaiodaaaa@361A@ j = 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.333 0.333 16.84 0.329 17.52 0.325 18.85
j = 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 4 0.667 0.667 18.68 0.672 18.70 0.678 18.89
j = 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 5 0.833 0.833 8.06 0.844 7.81 0.854 7.67
Table 4.3
Standard deviations of the sample sizes obtained with the cube method (CM) and with two rejective algorithms (R 5%, R 50%)
Table summary
This table displays the results of Standard deviations of the sample sizes obtained with the cube method (CM) and with two rejective algorithms (R 5% (équation) has a low variation and (équation) has a medium variation (appearing as column headers).
x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ has a low variation x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ has a medium variation
Sector Site x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ CM R 5 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ R 50 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ CM R 5 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ R 50 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@
i = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.000 0.894 1.371 3 0.000 0.891 1.371
j = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.000 0.854 1.295 2 0.000 0.831 1.294
j = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.000 0.896 1.377 3 0.000 0.891 1.374
i = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.130 0.828 1.293 1 0.144 0.654 1.013
j = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.195 0.832 1.298 2 0.170 0.831 1.290
j = 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0.179 0.826 1.296 2 0.141 0.830 1.297
i = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.339 0.859 1.366 2 0.342 0.835 1.294
j = 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 0.381 0.859 1.367 4 0.350 0.807 1.294
j = 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 4 0.319 0.822 1.288 5 0.248 0.655 1.010

In Tables 4.1 and 4.2, the cube method maintains the selection probabilities and yields a total estimator with the smallest standard deviations. Taking γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHZoWzdaahaaWcbeqaaiaaikdaaa aaaa@3415@ equal to the 50 th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaI1aGaaGimamaaCaaaleqabaGaae iDaiaabIgaaaaaaa@350D@ percentile of the χ 8 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHhpWydaqhaaWcbaGaaGioaaqaai aaikdaaaaaaa@34E7@ distribution for the rejective algorithm yields the poorer results, both in terms of selection probabilities and of the standard deviations of y ¯ s i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWG5bGbaebadaWgaaWcbaGaam4Cai aadMgacaWGQbaabeaakiaac6caaaa@3658@ The largest biases for the selection probabilities occur at the extreme x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4baaaa@3282@ values in Table 4.2. The selection probability for site j = 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaisdaaaa@33F9@ is underestimated by 11% with the rejective method based on the 50 th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaI1aGaaGimamaaCaaaleqabaGaae iDaiaabIgaaaaaaa@350D@ percentile and by 5% with the 5 th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaI1aWaaWbaaSqabeaacaqG0bGaae iAaaaaaaa@3453@ percentile. The probability is over estimated in the sites with the large values for x . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bGaaiOlaaaa@3334@

In Tables 4.1 and 4.2, the standard deviation for Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamyAai aadQgaaeqaaaaa@347C@ is, in most cases, smallest for the cube method and largest for the rejection algorithm based on the 50 th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaI1aGaaGimamaaCaaaleqabaGaae iDaiaabIgaaaaaaa@350D@ percentile. The standard deviations for the rejective algorithm are up to 10% larger than the ones for the cube method. In Table 4.2, the largest gain in efficiency of the cube method with respect to the R 5 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3557@ rejective algorithm (equal to the ratio of standard deviations squared) is 23%; it occurs when j = 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaisdaaaa@33F9@ and x = 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bGaaGypaiaaigdacaGGUaaaaa@34B6@ These standard deviations are driven by the variability in sample sizes n i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgacaWGQb aabeaakiaac6caaaa@353D@ Table 4.3 gives the sample sizes’ standard deviations. Since the expected number of visits to sector 1 and to sites 1, 2, and 3 are integers, the cube method is able to get sample sizes equal to their expectations for this sector and the sample sizes standard deviations are 0. This is not possible in sectors 2 and 3 as the expected sample sizes for these sectors are not integer valued. In general, the rejective algorithms give sample sizes whose standard deviations are much more variable than those for the cube method. This makes the rejective algorithm total estimators more variable than those obtained with the cube method.

The conditional variance estimator for fishing effort Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamyAai aadQgaaeqaaaaa@347C@ in site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ proposed in Section 3.2 is

v ( Y ^ i j ) = H 2 ( 1 n i j / H ) n i j h s i j ( y h i j y ¯ s i j ) 2 n i j 1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG2bWaaeWaaeaaceWGzbGbaKaada WgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaGaaGypamaa laaabaGaamisamaaCaaaleqabaGaaGOmaaaakmaabmaabaWaaSGbae aacaaIXaGaeyOeI0IaamOBamaaBaaaleaacaWGPbGaamOAaaqabaaa keaacaWGibaaaaGaayjkaiaawMcaaaqaaiaad6gadaWgaaWcbaGaam yAaiaadQgaaeqaaaaakmaaqafabaWaaSaaaeaadaqadaqaaiaadMha daWgaaWcbaGaamiAaiaadMgacaWGQbaabeaakiabgkHiTiqadMhaga qeamaaBaaaleaacaWGZbGaamyAaiaadQgaaeqaaaGccaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOBamaaBaaaleaacaWGPb GaamOAaaqabaGccqGHsislcaaIXaaaaaWcbaGaamiAaiabgIGiolaa dohadaWgaaadbaGaamyAaiaadQgaaeqaaaWcbeqdcqGHris5aOGaaG Olaaaa@5C69@

The conditional sampling properties, given n i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgacaWGQb aabeaakiaacYcaaaa@353B@ of this variance estimator were investigated in the Monte Carlo study with B = 10,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGcbGaaGypaiaabgdacaqGWaGaae ilaiaabcdacaqGWaGaaeimaaaa@3742@ balanced samples for the three sample designs. For each site and for each sample size n i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@3481@ the conditional variance Var ( Y ^ i j | n i j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGwbGaaeyyaiaabkhadaqadaqaam aaeiaabaGabmywayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaakiaa ykW7aiaawIa7aiaaykW7caWGUbWaaSbaaSqaaiaadMgacaWGQbaabe aaaOGaayjkaiaawMcaaaaa@4073@ and the conditional expectation of the variance estimator E { v ( Y ^ i j ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGfbWaaiWaaeaacaWG2bWaaeWaae aaceWGzbGbaKaadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGa ayzkaaaacaGL7bGaayzFaaaaaa@3A03@ were evaluated using the Monte Carlo samples for which the sample size for site ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@35A1@ was n i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgacaWGQb aabeaakiaac6caaaa@353D@ The conditional relative bias of the variance estimator, E { v ( Y ^ i j ) } / Var ( Y ^ i j | n i j ) 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaWcgaqaaiaabweadaGadaqaaiaadA hadaqadaqaaiqadMfagaqcamaaBaaaleaacaWGPbGaamOAaaqabaaa kiaawIcacaGLPaaaaiaawUhacaGL9baaaeaacaqGwbGaaeyyaiaabk hadaqadaqaamaaeiaabaGabmywayaajaWaaSbaaSqaaiaadMgacaWG QbaabeaakiaaykW7aiaawIa7aiaaykW7caWGUbWaaSbaaSqaaiaadM gacaWGQbaabeaaaOGaayjkaiaawMcaaiabgkHiTiaaigdaaaGaaiil aaaa@4B5F@ was then calculated. The conditional relative biases were then aggregated by weighting each sample size n i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@3481@ using its frequency in the 10,000 Monte Carlo samples; the results are in Table 5.1.

In Table 5.1, the aggregated relative biases are less than 3% in absolute value for the three selection algorithms. This validates the conditional variance estimator proposed in Section 3.2 for a single cell of the cross-classified table. The conditional variances of sums such as Y ^ i j + Y ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamyAai aadQgaaeqaaOGaey4kaSIabmywayaajaWaaSbaaSqaaiaadMgacaWG QbWaaWbaaWqabeaacWaGyBOmGikaaaWcbeaaaaa@3B78@ is more complicated as it involves joint selection probabilities; the estimation of these variances is not considered here. See Breidt and Chauvet (2011) for a discussion of variance estimation with the cube method.

Table 5.1
Aggregated conditional bias, in percentage, of the conditional variance estimator v ( Y ^ i j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG2bWaaeWaaeaaceWGzbGbaKaada WgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaaaaa@3704@ obtained with the cube method and two rejective algorithms (R 5%, R 50%)
Table summary
This table displays the results of Aggregated conditional bias x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ has a low variation and x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ has a medium variation (appearing as column headers).
x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ has a low variation x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ has a medium variation
Sector Site x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ CM R 5 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ R 50 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG4baaaa@34AF@ CM R 5 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@ R 50 % MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWGsbGaaGjbVlaaiwdacaaILaaaaa@3784@
i = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 1 -3 3 3 -1 1 1
j = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 2 -1 -2 2 3 1 -2
j = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 -1 0 1 3 0 -1 0
i = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 -2 2 0 1 1 -1 -2
j = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 1 -1 -1 2 2 2 3
j = 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 2 0 3 -2 2 0 0 -3
i = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdaaaa@3618@ j = 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 1 -3 2 2 0 -3 -1
j = 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 3 2 1 1 4 0 0 0
j = 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGypaiaaigdaaaa@3619@ 4 -1 1 -2 5 -2 -1 1

The conclusion of this Monte Carlo investigation is that the rejective algorithm changes the selection probabilities: sites with small importance are under represented in the rejective samples while the cube method is very good at preserving the selection probabilities. Under both algorithms the calibrated estimator for the total of y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFv0dc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0dYddrpe0db9Wqpepic9Lr=xfr=xfr=xmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5baaaa@3283@ in a domain is unbiased. Smaller variances are however obtained with the cube algorithm as it gives domain sample sizes that are less variable than the rejective algorithm.


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