Variance estimation under monotone non-response for a panel survey
Section 5. A simulation study

In this section, several artificial populations are generated according to the model described in Section 5.1. In Section 5.2, we consider several estimators for a change between totals, which illustrates the heuristic reasoning in Section 4. A Monte Carlo experiment is presented in Section 5.3, and several variance estimators for estimating a total, a ratio or a parameter change are compared. The results from Tables 5.1 and 5.2 are readily reproducible using the R code provided in the Supplementary Material.

5.1  Simulation set-up

We consider seven populations of size 10,000, each containing three variables of interest y i 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaaIXa aabeaakiaacYcaaaa@3559@ y i 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaaIYa aabeaaaaa@34A0@ and y i 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaaIZa aabeaaaaa@34A1@ observed at times t = 1, 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdacaaISaGaaG jbVlaaikdaaaa@3746@ and 3, respectively. The variables of interest are generated according to the superpopulation model

y i 1 = α 0 + α a x a i + α b x b i + σ u i 1 , ( 5.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaaIXa aabeaakiaai2dacqaHXoqydaahaaWcbeqaaiaaicdaaaGccqGHRaWk cqaHXoqydaahaaWcbeqaaiaadggaaaGccaWG4bWaaSbaaSqaaiaadg gacaWGPbaabeaakiabgUcaRiabeg7aHnaaCaaaleqabaGaamOyaaaa kiaadIhadaWgaaWcbaGaamOyaiaadMgaaeqaaOGaey4kaSIaeq4Wdm NaamyDamaaBaaaleaacaWGPbGaaGymaaqabaGccaaISaGaaGzbVlaa ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGynaiaac6cacaaIXaGaai ykaaaa@56CA@

y i 2 = ρ y i 1 + σ u i 2 , ( 5.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaaIYa aabeaakiaai2dacqaHbpGCcaWG5bWaaSbaaSqaaiaadMgacaaIXaaa beaakiabgUcaRiabeo8aZjaadwhadaWgaaWcbaGaamyAaiaaikdaae qaaOGaaGilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaa iwdacaGGUaGaaGOmaiaacMcaaaa@4B8E@

y i 3 = ρ y i 2 + σ u i 3 . ( 5.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaaIZa aabeaakiaai2dacqaHbpGCcaWG5bWaaSbaaSqaaiaadMgacaaIYaaa beaakiabgUcaRiabeo8aZjaadwhadaWgaaWcbaGaamyAaiaaiodaae qaaOGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaa iwdacaGGUaGaaG4maiaacMcaaaa@4B94@

The auxiliary variables x a i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadggacaWGPb aabeaaaaa@34C9@ and x b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadkgacaWGPb aabeaaaaa@34CA@ are independently generated from a Gamma distribution with shape and scale parameters 2 and 1. Two auxiliary variables x c i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadogacaWGPb aabeaaaaa@34CB@ and x d i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadsgacaWGPb aabeaakiaacYcaaaa@3586@ not related to the variables of interest, are generated similarly. The variables u i 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaaIXa aabeaakiaacYcaaaa@3555@ u i 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaaIYa aabeaaaaa@349C@ and u i 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaaIZa aabeaaaaa@349D@ are independently generated according to a standard normal distribution. We use α 0 = 10 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqydaahaaWcbeqaaiaaicdaaa GccaaI9aGaaGymaiaaicdacaGGSaaaaa@3748@ α a = α b = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqydaahaaWcbeqaaiaadggaaa GccaaI9aGaeqySde2aaWbaaSqabeaacaWGIbaaaOGaaGypaiaaiwda aaa@3992@ and σ = 10 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHdpWCcaaI9aGaaGymaiaaicdaca GGSaaaaa@367B@ which leads to a coefficient of determination ( R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadkfadaahaaWcbeqaai aaikdaaaaakiaawIcacaGLPaaaaaa@351F@ in model (5.1) approximately equal to 0.50. The parameter ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHbpGCaaa@338C@ is set to 0, 0.2, 0.4, 0.6, 0.8, 1.0 and 1.2 for populations 1 to 7, respectively.

For each population, a simple random sample s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaicdaaeqaaa aa@33AA@ of size n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbGaaGypaiaaykW7aaa@3511@  1,000 is selected. Three non-response phases are then successively simulated. At each phase δ = 1, 2, 3 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaaGOmaiaaiYcacaaMe8UaaG4maiaacYcaaaa@3BA2@ the sub-sample of respondents s δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiabes7aKbqaba aaaa@3495@ is obtained by Poisson sampling with a response probability p i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@3581@ for unit i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaiilaaaa@336A@ defined as

logit ( p i δ ) = β δ 0 + β δ a x a i + β δ b x b i . ( 5.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGSbGaae4BaiaabEgacaqGPbGaae iDamaabmaabaGaamiCamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGc caGLOaGaayzkaaGaaGypaiabek7aInaaCaaaleqabaGaeqiTdqMaaG imaaaakiabgUcaRiabek7aInaaCaaaleqabaGaeqiTdqMaamyyaaaa kiaadIhadaWgaaWcbaGaamyyaiaadMgaaeqaaOGaey4kaSIaeqOSdi 2aaWbaaSqabeaacqaH0oazcaWGIbaaaOGaamiEamaaBaaaleaacaWG IbGaamyAaaqabaGccaaIUaGaaGzbVlaaywW7caaMf8UaaGzbVlaayw W7caGGOaGaaGynaiaac6cacaaI0aGaaiykaaaa@5D5F@

We use β δ 0 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHYoGydaahaaWcbeqaaiabes7aKj aaicdaaaGccaaI9aGaeyOeI0IaaGymaaaa@3872@ at each phase δ = 1, 2, 3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaaGOmaiaaiYcacaaMe8UaaG4maiaac6caaaa@3BA4@ For δ = 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaacYcaaa a@35A3@ we use β 1 a = β 1 b = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHYoGydaahaaWcbeqaaiaaigdaca WGHbaaaOGaaGypaiabek7aInaaCaaaleqabaGaaGymaiaadkgaaaGc caaI9aGaaGPaVdaa@3BD8@  0.60, which corresponds to an average response rate of 0.75. For δ = 2, 3 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGOmaiaaiYcaca aMe8UaaG4maiaacYcaaaa@38A4@ we use β δ a = β δ b = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHYoGydaahaaWcbeqaaiabes7aKj aadggaaaGccaaI9aGaeqOSdi2aaWbaaSqabeaacqaH0oazcaWGIbaa aOGaaGypaiaaykW7aaa@3DAC@ 0.75, which corresponds to an average response rate of 0.81. Inside each sub-sample s δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiabes7aKbqaba GccaGGSaaaaa@354F@ the estimated response probabilities p ^ i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaaaaa@3591@ are obtained by means of an unweighted logistic regression.

5.2  Comparison of estimators for a difference of totals

In this section, we are interested in comparing the accuracy of two estimators for a difference of totals Δ ( u t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqqHuoardaqadaqaaiaadwhacqGHsg IRcaWG0baacaGLOaGaayzkaaaaaa@389B@ for u = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaigdaaaa@3448@ and t = 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaikdacaGGSaaaaa@34F8@ for u = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaigdaaaa@3448@ and t = 3 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaiodacaGGSaaaaa@34F9@ and for u = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaikdaaaa@3449@ and t = 3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaiodacaGGUaaaaa@34FB@ We consider the estimator Δ ^ u t ( u t ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG1b GaamiDaaqabaGcdaqadaqaaiaadwhacqGHsgIRcaWG0baacaGLOaGa ayzkaaGaaiilaaaa@3B84@ which makes use of the whole appropriate sub-samples for variables y i u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG1b aabeaaaaa@34DE@ and y i t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaakiaacYcaaaa@3597@ and the estimator Δ ^ t t ( u t ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaaqabaGcdaqadaqaaiaadwhacqGHsgIRcaWG0baacaGLOaGa ayzkaaGaaiilaaaa@3B83@ which makes use of the common sub-sample only. These two estimators are compared through the relative difference (RD) of their variances, which are defined as follows:

RD ( u t ) = 100 × V { Δ ^ u t ( u t ) } V { Δ ^ t t ( u t ) } V { Δ ^ t t ( u t ) } . ( 5.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGsbGaaeiramaabmaabaGaamyDai abgkziUkaadshaaiaawIcacaGLPaaacaaI9aGaaGymaiaaicdacaaI WaGaey41aq7aaSaaaeaacaWGwbWaaiWaaeaacaaMc8UafuiLdqKbaK aadaWgaaWcbaGaamyDaiaadshaaeqaaOWaaeWaaeaacaWG1bGaeyOK H4QaamiDaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiabgkHiTiaadA fadaGadaqaaiaaykW7cuqHuoargaqcamaaBaaaleaacaWG0bGaamiD aaqabaGcdaqadaqaaiaadwhacqGHsgIRcaWG0baacaGLOaGaayzkaa aacaGL7bGaayzFaaaabaGaamOvamaacmaabaGaaGPaVlqbfs5aezaa jaWaaSbaaSqaaiaadshacaWG0baabeaakmaabmaabaGaamyDaiabgk ziUkaadshaaiaawIcacaGLPaaaaiaawUhacaGL9baaaaGaaGOlaiaa ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGynaiaac6cacaaI1aGaai ykaaaa@722E@

The true variances are replaced by their Monte Carlo approximation, obtained by repeating B = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGcbGaaGypaiaaykW7aaa@34E5@  100,000 times the sample selection and the non-response phases.

The results are presented in Table 5.1. A positive RD indicates that the use of the common sample only leads to a more accurate estimator. As could be expected, the RD increases in all cases with ρ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHbpGCcaGGSaaaaa@343C@ that is, when the correlation between y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ and y i u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG1b aabeaaaaa@34DE@ increases. For u = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaigdaaaa@3448@ and t = 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaikdacaGGSaaaaa@34F8@ and for u = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaikdaaaa@3449@ and t = 3 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaiodacaGGSaaaaa@34F9@ the estimator Δ ^ t t ( u t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaaqabaGcdaqadaqaaiaadwhacqGHsgIRcaWG0baacaGLOaGa ayzkaaaaaa@3AD3@ is more accurate for ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHbpGCaaa@338C@ greater than 0.6. For u = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaigdaaaa@3448@ and t = 3 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaiodacaGGSaaaaa@34F9@ Δ ^ t t ( u t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaaqabaGcdaqadaqaaiaadwhacqGHsgIRcaWG0baacaGLOaGa ayzkaaaaaa@3AD3@ is more accurate for ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHbpGCaaa@338C@ greater than 0.8.

Table 5.1
Relative Difference (RD) between two estimators for a difference of totals
Table summary
This table displays the results of Relative Difference (RD) between two estimators for a difference of totals. The information is grouped by ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacqaHbpGCaaa@35B9@ (appearing as row headers), RD( 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGsbGaaeiramaabmaabaGaaGymai abgkziUkaaikdaaiaawIcacaGLPaaaaaa@3A82@ , RD( 13 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGsbGaaeiramaabmaabaGaaGymai abgkziUkaaiodaaiaawIcacaGLPaaaaaa@3A83@ and RD( 23 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGsbGaaeiramaabmaabaGaaGOmai abgkziUkaaiodaaiaawIcacaGLPaaaaaa@3A84@ (appearing as column headers).
ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacqaHbpGCaaa@35B9@ RD( 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGsbGaaeiramaabmaabaGaaGymai abgkziUkaaikdaaiaawIcacaGLPaaaaaa@3A82@ RD( 13 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGsbGaaeiramaabmaabaGaaGymai abgkziUkaaiodaaiaawIcacaGLPaaaaaa@3A83@ RD( 23 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaqGsbGaaeiramaabmaabaGaaGOmai abgkziUkaaiodaaiaawIcacaGLPaaaaaa@3A84@
0.0 -12 -27 -13
0.2 -09 -25 -11
0.4 -04 -20 -03
0.6 -05 -09 11
0.8 17 11 39
1.0 30 33 83
1.2 40 46 127

5.3  Performances of the variance estimators

In this section, we consider the artificial population 5 ( ρ = 0 .8 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiabeg8aYjaai2dacaqGWa GaaeOlaiaabIdaaiaawIcacaGLPaaaaaa@37FB@ generated as described in Section 5.1. The sample selection by means of simple random sampling of size n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbGaaGypaiaaykW7aaa@3511@  1,000 and the three non-response phases are applied B = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGcbGaaGypaiaaykW7aaa@34E5@  5,000 times. We are interested in evaluating the variance estimators and the simplified variance estimators, in case of estimating a total, a ratio or a change in totals.

As for the total Y ( t ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGzbWaaeWaaeaacaWG0baacaGLOa GaayzkaaGaaiilaaaa@35DC@ we consider at each time t = 1, 2, 3, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdacaaISaGaaG jbVlaaikdacaaISaGaaGjbVlaaiodacaaISaaaaa@3AFC@ three estimators. The estimator Y ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@33DF@ makes use of the weights d t i = π i 1 ( p ^ i 1 t ) 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGKbWaaSbaaSqaaiaadshacaWGPb aabeaakiaai2dacqaHapaCdaqhaaWcbaGaamyAaaqaaiabgkHiTiaa igdaaaGcdaqadaqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaG ymaiaaykW7cqGHsgIRcaaMc8UaamiDaaaaaOGaayjkaiaawMcaamaa CaaaleqabaGaeyOeI0IaaGymaaaakiaaygW7caGGUaaaaa@48A8@ The estimator Y ^ w t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaaaa@34DB@ makes use of the weights w i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgaaeqaaO Gaaiilaaaa@349C@ obtained by calibrating the weights d t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGKbWaaSbaaSqaaiaadshacaWGPb aabeaaaaa@34C8@ on the population size and on the totals of the auxiliary variables x a i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadggacaWGPb aabeaaaaa@34C9@ and x b i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadkgacaWGPb aabeaakiaac6caaaa@3586@ The estimator Y ^ w ˜ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGabm4Day aaiaGaamiDaaqabaaaaa@34EA@ makes use of the weights w ˜ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWG3bGbaGaadaWgaaWcbaGaamyAaa qabaGccaGGSaaaaa@34AB@ obtained by calibrating the weights d t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGKbWaaSbaaSqaaiaadshacaWGPb aabeaaaaa@34C8@ on the population size and on the totals of the auxiliary variables x c i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadogacaWGPb aabeaaaaa@34CB@ and x d i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadsgacaWGPb aabeaakiaac6caaaa@3588@ The working model is therefore well-specified for Y ^ w t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaOGaaiilaaaa@3595@ but not for Y ^ w ˜ t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGabm4Day aaiaGaamiDaaqabaGccaGGUaaaaa@35A6@ The proposed variance estimator for Y ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@33DF@ is obtained from equation (2.16), and the simplified variance estimator is obtained by plugging in (2.16) the simplified variance estimator for non-response given in (2.17). The proposed variance estimators for Y ^ w t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaaaa@34DB@ and Y ^ w ˜ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGabm4Day aaiaGaamiDaaqabaaaaa@34EA@ are obtained from equation (3.8), and the simplified variance estimators are obtained by plugging in (3.8) the simplified variance estimator for non-response given in (3.9).

We are also interested in estimating the ratio R ( t ) = Y ( t ) / Y ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbWaaeWaaeaacaWG0baacaGLOa GaayzkaaGaaGypamaalyaabaGaamywamaabmaabaGaamiDaaGaayjk aiaawMcaaaqaaiaadMfadaqadaqaaiaaigdaaiaawIcacaGLPaaaaa aaaa@3C84@ for t = 2, 3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaikdacaaISaGaaG jbVlaaiodacaGGUaaaaa@37FA@ At each time t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiilaaaa@3375@ we consider three estimators. The estimator R ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGsbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@33D8@ makes use of the weights d i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGKbWaaSbaaSqaaiaadMgaaeqaaO GaaiOlaaaa@348B@ The proposed variance estimator is obtained from equation (3.14), by using the estimated linearized variable u i t = ( Y ^ 1 ) 1 ( y t i R ^ t y 1 i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaWG0b aabeaakiaai2dadaqadaqaaiqadMfagaqcamaaBaaaleaacaaIXaaa beaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakm aabmaabaGaamyEamaaBaaaleaacaWG0bGaamyAaaqabaGccqGHsisl ceWGsbGbaKaadaWgaaWcbaGaamiDaaqabaGccaWG5bWaaSbaaSqaai aaigdacaWGPbaabeaaaOGaayjkaiaawMcaaiaac6caaaa@4627@ The simplified variance estimator is obtained by plugging in (3.14) the simplified variance estimator for non-response given in (3.15). The estimators R ^ w t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGsbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaaaa@34D4@ and R ^ w ˜ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGsbGbaKaadaWgaaWcbaGabm4Day aaiaGaamiDaaqabaaaaa@34E3@ make use of the calibrated weights w i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgaaeqaaa aa@33E2@ and w ˜ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWG3bGbaGaadaWgaaWcbaGaamyAaa qabaGccaGGUaaaaa@34AD@ The proposed variance estimators are obtained from equation (3.21). The simplified variance estimators are obtained by plugging in (3.21) the simplified variance estimator for non-response given in (3.22).

Finally, we are interested in estimating the change in totals Δ ( 1 t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqqHuoardaqadaqaaiaaigdacqGHsg IRcaWG0baacaGLOaGaayzkaaaaaa@385C@ for t = 2, 3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaikdacaaISaGaaG jbVlaaiodacaGGUaaaaa@37FA@ At each time t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiilaaaa@3375@ we consider three estimators. The estimator Δ ^ t t ( 1 t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaaqabaGcdaqadaqaaiaaigdacqGHsgIRcaWG0baacaGLOaGa ayzkaaaaaa@3A94@ makes use of the weights d i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGKbWaaSbaaSqaaiaadMgaaeqaaO GaaiOlaaaa@348B@ The proposed variance estimator is obtained from equation (4.8), and the simplified variance estimator is obtained by plugging in (4.8) the simplified variance estimator for non-response given in (4.9). The estimators Δ ^ t t , w ( 1 t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaiaaiYcacaaMe8Uaam4DaaqabaGcdaqadaqaaiaaigdacqGH sgIRcaWG0baacaGLOaGaayzkaaaaaa@3DD3@ and Δ ^ t t , w ˜ ( 1 t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaiaaiYcacaaMe8Uabm4DayaaiaaabeaakmaabmaabaGaaGym aiabgkziUkaadshaaiaawIcacaGLPaaaaaa@3DE2@ make use of the calibrated weights w i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgaaeqaaa aa@33E2@ and w ˜ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWG3bGbaGaadaWgaaWcbaGaamyAaa qabaGccaGGUaaaaa@34AD@ The proposed variance estimators are obtained from equation (4.8), by replacing y i t y i u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaakiabgkHiTiaadMhadaWgaaWcbaGaamyAaiaadwhaaeqaaaaa @38E6@ by the estimated residual for the weighted regression of y i t y i u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaakiabgkHiTiaadMhadaWgaaWcbaGaamyAaiaadwhaaeqaaaaa @38E6@ on the calibration variables. The simplified variance estimators are obtained by plugging in (4.8) the simplified variance estimator for non-response given in (4.9).

For a proposed variance estimator V ^ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaacaGGSaaaaa@3367@ we computed the Monte Carlo Percent Relative Bias

RB mc ( V ^ ) = 100 × B 1 b = 1 B V ^ ( b ) V V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGsbGaaeOqamaaBaaaleaacaqGTb Gaae4yaaqabaGcdaqadaqaaiqadAfagaqcaaGaayjkaiaawMcaaiaa i2dacaaIXaGaaGimaiaaicdacqGHxdaTdaWcaaqaaiaadkeadaahaa WcbeqaaiabgkHiTiaaigdaaaGcdaaeWaqaaiqadAfagaqcamaaCaaa leqabaWaaeWaaeaacaWGIbaacaGLOaGaayzkaaaaaaqaaiaadkgaca aI9aGaaGymaaqaaiaadkeaa0GaeyyeIuoakiabgkHiTiaadAfaaeaa caWGwbaaaaaa@4B04@

where the global variance V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbaaaa@32A7@ was approximated through an independent set of 100,000 simulations. To evaluate the contribution of some component V ^ a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaWgaaWcbaGaamyyaa qabaaaaa@33C9@ into the variance estimator V ^ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaacaGGSaaaaa@3367@ we computed the contribution (in percent)

CONTR mc ( V ^ a ) = 100 × 1 B b = 1 B V ^ a ( b ) 1 B b = 1 B V ^ ( b ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGdbGaae4taiaab6eacaqGubGaae OuamaaBaaaleaacaqGTbGaae4yaaqabaGcdaqadaqaaiqadAfagaqc amaaBaaaleaacaWGHbaabeaaaOGaayjkaiaawMcaaiaai2dacaaIXa GaaGimaiaaicdacqGHxdaTdaWcaaqaamaaleaaleaacaaIXaaabaGa amOqaaaakmaaqadabaGabmOvayaajaWaa0baaSqaaiaadggaaeaada qadaqaaiaadkgaaiaawIcacaGLPaaaaaaabaGaamOyaiaai2dacaaI XaaabaGaamOqaaqdcqGHris5aaGcbaWaaSqaaSqaaiaaigdaaeaaca WGcbaaaOWaaabmaeaaceWGwbGbaKaadaahaaWcbeqaamaabmaabaGa amOyaaGaayjkaiaawMcaaaaaaeaacaWGIbGaaGypaiaaigdaaeaaca WGcbaaniabggHiLdaaaOGaaGOlaaaa@56F9@

To evaluate the simplified variance estimator for the non-response V ^ simp nr , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaae4Cai aabMgacaqGTbGaaeiCaaqaaiaab6gacaqGYbaaaOGaaiilaaaa@3949@ we computed the Monte Carlo Percent Relative Bias

RB mc ( V ^ simp nr ) = 100 × B 1 b = 1 B V ^ simp ( b ) V nr V nr , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGsbGaaeOqamaaBaaaleaacaqGTb Gaae4yaaqabaGcdaqadaqaaiqadAfagaqcamaaDaaaleaacaqGZbGa aeyAaiaab2gacaqGWbaabaGaaeOBaiaabkhaaaaakiaawIcacaGLPa aacaaI9aGaaGymaiaaicdacaaIWaGaey41aq7aaSaaaeaacaWGcbWa aWbaaSqabeaacqGHsislcaaIXaaaaOWaaabmaeaaceWGwbGbaKaada qhaaWcbaGaae4CaiaabMgacaqGTbGaaeiCaaqaamaabmaabaGaamOy aaGaayjkaiaawMcaaaaaaeaacaWGIbGaaGypaiaaigdaaeaacaWGcb aaniabggHiLdGccqGHsislcaWGwbWaaWbaaSqabeaacaqGUbGaaeOC aaaaaOqaaiaadAfadaahaaWcbeqaaiaab6gacaqGYbaaaaaakiaaiY caaaa@599B@

where the variance V nr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaqGUbGaae OCaaaaaaa@34BA@ due to non-response was approximated through an independent set of 100,000 simulations.

The simulation results are presented in Table 5.2. The proposed variance estimator is almost unbiased in all cases. As could be expected, the contribution of the variance due to the sampling design decreases with time, as the number of respondents decreases and as the variance due to non-response becomes larger. The simplified variance estimator is highly biased for the variance due to non-response in case of Y ^ t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaGccaGGUaaaaa@349B@ The bias decreases quickly with time, but remains large at time t = 3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaiodacaGGUaaaaa@34FB@ The simplified variance estimator is almost unbiased for a calibrated estimator when the working model is adequately specified, but is severely biased otherwise. This is consistent with our reasoning in Section 3.1. The simplified variance estimator is almost unbiased for the three estimators of the ratio, and for the calibrated estimators of the change in totals. In case of the non-calibrated estimator for the change in totals, the bias can be as high as 30%.

Table 5.2
Relative bias of a global variance estimator, relative contribution to the estimators of variance components and relative bias of a simplified variance estimator for the variance due to non-response for the estimation of a total, a ratio or a change in totals with three sets of weights
Table summary
This table displays the results of Relative bias of a global variance estimator. The information is grouped by (appearing as row headers), (équation) (appearing as column headers).
t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@ t=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWG0bGaaGypaiaaigdaaaa@3674@
Y ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFj0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@3600@ Y ^ wt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFj0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaaaa@36FC@ Y ^ w ˜ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFj0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGabm4Day aaiaGaamiDaaqabaaaaa@370B@ R ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGsbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@35FB@ R ^ wt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGsbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaaaa@36F7@ R ^ w ˜ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGsbGbaKaadaWgaaWcbaGabm4Day aaiaGaamiDaaqabaaaaa@3706@ Δ ^ tt ( 1t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaaqabaGcdaqadaqaaiaaigdacqGHsgIRcaWG0baacaGLOaGa ayzkaaaaaa@3CB7@ Δ ^ tt,w ( 1t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaiaaiYcacaaMc8Uaam4DaaqabaGcdaqadaqaaiaaigdacqGH sgIRcaWG0baacaGLOaGaayzkaaaaaa@3FF4@ Δ ^ tt, w ˜ ( 1t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuqHuoargaqcamaaBaaaleaacaWG0b GaamiDaiaaiYcacaaMc8Uabm4DayaaiaaabeaakmaabmaabaGaaGym aiabgkziUkaadshaaiaawIcacaGLPaaaaaa@4003@
RB mc ( V ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGsbGaaeOqamaaBaaaleaacaqGTb Gaae4yaaqabaGcdaqadaqaaiqadAfagaqcaaGaayjkaiaawMcaaaaa @3A09@ 0 -1 -2 -1 -1 -2 -1 -1 -3 - 0 -2 - -1 -2 - -1 -2 - 0 -2 - 0 -2 - -1 -3
CONTR mc ( V ^ t p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGdbGaae4taiaab6eacaqGubGaae OuamaaBaaaleaacaqGTbGaae4yaaqabaGcdaqadaqaaiqadAfagaqc amaaDaaaleaacaWG0baabaGaamiCaaaaaOGaayjkaiaawMcaaaaa@3EA9@ 81 57 35 69 49 32 80 56 35 - 49 32 - 49 32 - 50 33 - 50 33 - 49 32 - 50 33
CONTR mc ( V ^ t nr1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGdbGaae4taiaab6eacaqGubGaae OuamaaBaaaleaacaqGTbGaae4yaaqabaGcdaqadaqaaiqadAfagaqc amaaDaaaleaacaWG0baabaGaaeOBaiaabkhacaaIXaaaaaGccaGLOa Gaayzkaaaaaa@4055@ 19 19 13 31 22 15 20 18 13 - 22 15 - 22 15 - 22 15 - 22 14 - 22 15 - 22 14
CONTR mc ( V ^ t nr2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGdbGaae4taiaab6eacaqGubGaae OuamaaBaaaleaacaqGTbGaae4yaaqabaGcdaqadaqaaiqadAfagaqc amaaDaaaleaacaWG0baabaGaaeOBaiaabkhacaaIYaaaaaGccaGLOa Gaayzkaaaaaa@4056@ - 25 18 - 28 19 - 25 17 - 28 19 - 28 19 - 28 19 - 28 18 - 28 19 - 28 18
CONTR mc ( V ^ t nr3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGdbGaae4taiaab6eacaqGubGaae OuamaaBaaaleaacaqGTbGaae4yaaqabaGcdaqadaqaaiqadAfagaqc amaaDaaaleaacaWG0baabaGaaeOBaiaabkhacaaIZaaaaaGccaGLOa Gaayzkaaaaaa@4057@ - - 34 - - 34 - - 34 - - 34 - - 34 - - 34 - - 34 - - 34 - - 34
RB mc ( V ^ t,simp nr ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGsbGaaeOqamaaBaaaleaacaqGTb Gaae4yaaqabaGcdaqadaqaaiqadAfagaqcamaaDaaaleaacaWG0bGa aGilaiaaysW7caqGZbGaaeyAaiaab2gacaqGWbaabaGaaeOBaiaabk haaaaakiaawIcacaGLPaaaaaa@4327@ 559 188 80 0 -1 -2 83 34 15 - 0 0 - -1 -2 - -1 -1 - 19 30 - -1 -2 - 3 5

Report a problem on this page

Is something not working? Is there information outdated? Can't find what you're looking for?

Please contact us and let us know how we can help you.

Privacy notice

Date modified: