Variance estimation under monotone non-response for a panel survey
Section 2. Correction of non-response and attrition

2.1  Notation and main assumptions

We are interested in a finite population U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbGaaiOlaaaa@3358@ A sample s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaicdaaeqaaa aa@33AA@ is first selected according to some sampling design p ( ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaacqGHflY1aiaawI cacaGLPaaacaGGSaaaaa@3744@ and we assume that the first-order inclusion probabilities π i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqaba aaaa@34A3@ are strictly positive for any i U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyicI4Saamyvaiaac6caaa a@35CA@ This first sampling phase corresponds to the original inclusion of units in the sample.

We consider the case of a panel survey in which the sole units in the original sample s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaicdaaeqaaa aa@33AA@ are followed over time, without reentry or late entry units at subsequent times to represent possible newborns. We are therefore interested in estimating some parameter defined over the population U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbGaaiilaaaa@3356@ for some study variable y t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadshaaeqaaa aa@33EF@ taking the value y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ for the unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32BA@ at time t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiOlaaaa@3377@ The units in the sample s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaicdaaeqaaa aa@33AA@ are followed at subsequent times δ = 1, , t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0bGaaiilaaaa@3C44@ and the sample is prone to unit non-response at each time. We note r i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGYbWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@3583@ for the response indicator for unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32BA@ at time δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaGGSaaaaa@3421@ and s δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiabes7aKbqaba aaaa@3495@ for the subset of respondents at time δ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaGGUaaaaa@3423@

We assume monotone non-response resulting in the nested sequence s 0 s 1 ... s t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaicdaaeqaaO Gaey4GIKSaam4CamaaBaaaleaacaaIXaaabeaakiabgoOijlaai6ca caaIUaGaaGOlaiabgoOijlaadohadaWgaaWcbaGaamiDaaqabaGcca GGUaaaaa@408C@ For δ = 1, , t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0bGaaiilaaaa@3C44@ we note p i δ = Pr ( i s δ | s δ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaI9aGaaeiuaiaabkhadaqadaqaaiaadMgacqGHiiIZ caWGZbWaaSbaaSqaaiabes7aKbqabaGcdaabbaqaaiaaykW7caWGZb WaaSbaaSqaaiabes7aKjabgkHiTiaaigdaaeqaaaGccaGLhWoaaiaa wIcacaGLPaaaaaa@4682@ for the response probability of some unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32BA@ to be a respondent at time δ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaGGUaaaaa@3423@ We assume that the data are missing at random, i.e. the response probability p i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@3581@ at time δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazaaa@3371@ can be explained by the variables observed at times 0, , δ 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIWaGaaGilaiaaysW7cqWIMaYsca aISaGaaGjbVlabes7aKjabgkHiTiaaigdacaGGSaaaaa@3C2B@ including the variables of interest, see for example Zhou and Kim (2012). Also, we assume that at any time δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazaaa@3371@ the units answer independently of one another, and we note p i j δ = p i δ p j δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaa0baaSqaaiaadMgacaWGQb aabaGaeqiTdqgaaOGaaGypaiaadchadaqhaaWcbaGaamyAaaqaaiab es7aKbaakiaadchadaqhaaWcbaGaamOAaaqaaiabes7aKbaaaaa@3EB6@ for the probability that two distinct units i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32BA@ and j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbaaaa@32BB@ answer jointly at time δ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaGGUaaaaa@3423@

2.2  Reweighted estimator

We are interested in estimating the total Y ( t ) = i U y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGzbWaaeWaaeaacaWG0baacaGLOa GaayzkaaGaaGypamaaqababeWcbaGaamyAaiabgIGiolaadwfaaeqa niabggHiLdGccaaMe8UaamyEamaaBaaaleaacaWGPbGaamiDaaqaba aaaa@3FCB@ at time t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiOlaaaa@3377@ In practice, the response probabilities at each time are unknown and need to be estimated. We assume that at each time δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazaaa@3371@ the probability of response is parametrically modeled as

p i δ = f δ ( z i δ , α δ ) ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaI9aGaamOzamaaCaaaleqabaGaeqiTdqgaaOWaaeWa aeaacaWG6bWaa0baaSqaaiaadMgaaeaacqaH0oazaaGccaaMb8UaaG ilaiaaysW7cqaHXoqydaahaaWcbeqaaiabes7aKbaaaOGaayjkaiaa wMcaaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaikdaca GGUaGaaGymaiaacMcaaaa@50FA@

for some known function f δ ( , ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGMbWaaWbaaSqabeaacqaH0oazaa GcdaqadaqaaiabgwSixlaaiYcacqGHflY1aiaawIcacaGLPaaacaGG Saaaaa@3C16@ where z i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG6bWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@358B@ is a vector of variables observed for all the units in s δ 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiabes7aKjabgk HiTiaaigdaaeqaaOGaaiilaaaa@36F7@ and α δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqydaahaaWcbeqaaiabes7aKb aaaaa@353D@ denotes some unknown parameter. Here and elsewhere, the superscript δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazaaa@3371@ will be used when we account for non-response at time δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaGGSaaaaa@3421@ like for the probability p i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@3581@ of unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32BA@ to be a respondent at time δ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaGGUaaaaa@3423@ Following the approach in Kim and Kim (2007), we assume that the true parameter is estimated by α ^ δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHXoqygaqcamaaCaaaleqabaGaeq iTdqgaaOGaaGzaVlaacYcaaaa@3791@ the solution of the estimating equation

α i s δ 1 k i δ { r i δ ln ( p i δ ) + ( 1 r i δ ) ln ( 1 p i δ ) } = 0, ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaWcaaqaaiabgkGi2cqaaiabgkGi2k abeg7aHbaadaaeqbqabSqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqa aiabes7aKjabgkHiTiaaigdaaeqaaaWcbeqdcqGHris5aOGaaGPaVl aadUgadaqhaaWcbaGaamyAaaqaaiabes7aKbaakmaacmaabaGaaGPa VlaadkhadaqhaaWcbaGaamyAaaqaaiabes7aKbaakiGacYgacaGGUb WaaeWaaeaacaWGWbWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaa wIcacaGLPaaacqGHRaWkdaqadaqaaiaaigdacqGHsislcaWGYbWaa0 baaSqaaiaadMgaaeaacqaH0oazaaaakiaawIcacaGLPaaaciGGSbGa aiOBamaabmaabaGaaGymaiabgkHiTiaadchadaqhaaWcbaGaamyAaa qaaiabes7aKbaaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaiabg2da 9iaaicdacaaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOa GaaGOmaiaac6cacaaIYaGaaiykaaaa@71B8@

with k i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@357C@ some weight of unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32BA@ in the estimating equation. Customary choices for these weights include k i δ = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaI9aGaaGymaaaa@3708@ and k i δ = π i 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaI9aGaeqiWda3aa0baaSqaaiaadMgaaeaacqGHsisl caaIXaaaaOGaaGzaVlaacYcaaaa@3D11@ see Fuller and An (1998), Beaumont (2005) and Kim and Kim (2007).

The estimated response probability at time δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazaaa@3371@ is p ^ i δ = f δ ( z i δ , α ^ δ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaakiaai2dacaWGMbWaaWbaaSqabeaacqaH0oazaaGc daqadaqaaiaadQhadaqhaaWcbaGaamyAaaqaaiabes7aKbaakiaayg W7caaISaGaaGjbVlqbeg7aHzaajaWaaWbaaSqabeaacqaH0oazaaaa kiaawIcacaGLPaaacaGGUaaaaa@4685@ The propensity score adjusted estimator at time t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiilaaaa@3375@ which will be simply called the reweighted estimator in what follows, is defined as

Y ^ t ( t ) = i s t y i t π i p ^ i 1 t with p ^ i 1 t = δ = 1 t p ^ i δ . ( 2.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaacaaI9aWaaabuaeqa leaacaWGPbGaeyicI4Saam4CamaaBaaameaacaWG0baabeaaaSqab0 GaeyyeIuoakiaaysW7daWcaaqaaiaadMhadaWgaaWcbaGaamyAaiaa dshaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGabmiCay aajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkziUkaaykW7 caWG0baaaaaakiaaywW7caaMf8Uaae4DaiaabMgacaqG0bGaaeiAai aaywW7caaMf8UabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGa aGPaVlabgkziUkaaykW7caWG0baaaOGaaGypamaarahabeWcbaGaeq iTdqMaaGypaiaaigdaaeaacaWG0baaniabg+GivdGccaaMe8UabmiC ayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazaaGccaaIUaGaaGzbVl aaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaIZaGa aiykaaaa@79D6@

Here and elsewhere, the subscript t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0baaaa@32C5@ will be used when the sample observed at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0baaaa@32C5@ is used for estimation, like for Y ^ t ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaGcdaqadaqaaiabgwSixdGaayjkaiaawMcaaaaa@37BC@ which makes use of the sample s t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaadshaaeqaaO GaaiOlaaaa@34A5@ We simplify the notation as Y ^ t ( t ) Y ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaacaaMe8UaaGjbVlab ggMi6kaaysW7caaMe8UabmywayaajaWaaSbaaSqaaiaadshaaeqaaa aa@407B@ when the total at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0baaaa@32C5@ is estimated by using the sample observed at time t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiOlaaaa@3377@

2.3  Variance computation

Under some regularity assumptions on the response mechanisms and some regularity conditions on the p δ ( , ) s, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaWbaaSqabeaacqaH0oazaa GcdaqadaqaaiabgwSixlaaiYcacqGHflY1aiaawIcacaGLPaaaieaa caWFzaIaae4CaiaabYcaaaa@3DD8@ we obtain from Theorem 1 in Kim and Kim (2007) that we can write

Y ^ t = Y ^ lin , t ( t ) + O p ( N n 1 ) , ( 2.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaGccaaI9aGabmywayaajaWaaSbaaSqaaiaabYgacaqGPbGaaeOB aiaaygW7caaISaGaaGjbVlaadshaaeqaaOWaaeWaaeaacaWG0baaca GLOaGaayzkaaGaey4kaSIaam4tamaaBaaaleaacaWGWbaabeaakmaa bmaabaGaamOtaiaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaaaki aawIcacaGLPaaacaaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7 caGGOaGaaGOmaiaac6cacaaI0aGaaiykaaaa@53F7@

where

Y ^ lin , t ( t ) = i s t 1 1 π i p ^ i 1 t 1 { k i t π i p ^ i 1 t 1 p i t ( h i t ) γ t + r i t p i t ( y i t k i t π i p ^ i 1 t 1 p i t ( h i t ) γ t ) } , ( 2.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaaeiBai aabMgacaqGUbGaaGzaVlaaiYcacaaMe8UaamiDaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaacaaI9aWaaabuaeqaleaacaWGPbGaey icI4Saam4CamaaBaaameaacaWG0bGaeyOeI0IaaGymaaqabaaaleqa niabggHiLdGccaaMe8+aaSaaaeaacaaIXaaabaGaeqiWda3aaSbaaS qaaiaadMgaaeqaaOGabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaI XaGaaGPaVlabgkziUkaaykW7caWG0bGaaGPaVlabgkHiTiaaykW7ca aIXaaaaaaakiaaysW7daGadaqaaiaadUgadaqhaaWcbaGaamyAaaqa aiaadshaaaGccqaHapaCdaWgaaWcbaGaamyAaaqabaGcceWGWbGbaK aadaqhaaWcbaGaamyAaaqaaiaaigdacaaMc8UaeyOKH4QaaGPaVlaa dshacaaMc8UaeyOeI0IaaGPaVlaaigdaaaGccaWGWbWaa0baaSqaai aadMgaaeaacaWG0baaaOWaaeWaaeaacaWGObWaa0baaSqaaiaadMga aeaacaWG0baaaaGccaGLOaGaayzkaaWaaWbaaSqabeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGabbiab=rQivcaakiabeo7a NnaaCaaaleqabaGaamiDaaaakiabgUcaRmaalaaabaGaamOCamaaDa aaleaacaWGPbaabaGaamiDaaaaaOqaaiaadchadaqhaaWcbaGaamyA aaqaaiaadshaaaaaaOGaaGjbVpaabmaabaGaamyEamaaBaaaleaaca WGPbGaamiDaaqabaGccqGHsislcaWGRbWaa0baaSqaaiaadMgaaeaa caWG0baaaOGaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGabmiCayaaja Waa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkziUkaaykW7caWG 0bGaaGPaVlabgkHiTiaaykW7caaIXaaaaOGaamiCamaaDaaaleaaca WGPbaabaGaamiDaaaakmaabmaabaGaamiAamaaDaaaleaacaWGPbaa baGaamiDaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGae8hPIujaaO Gaeq4SdC2aaWbaaSqabeaacaWG0baaaaGccaGLOaGaayzkaaaacaGL 7bGaayzFaaGaaiilaiaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaai OlaiaaiwdacaGGPaaaaa@BDB9@

and where for any δ = 1, , t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0baaaa@3B94@ we denote by h i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGObWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@3579@ the value of h i δ ( α ) = logit ( p i δ ) / α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGObWaa0baaSqaaiaadMgaaeaacq aH0oazaaGcdaqadaqaaiabeg7aHbGaayjkaiaawMcaaiaai2dadaWc gaqaaiabgkGi2kaabYgacaqGVbGaae4zaiaabMgacaqG0bWaaeWaae aacaWGWbWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaawIcacaGL PaaaaeaacqGHciITcqaHXoqyaaaaaa@47E9@ evaluated at α = α δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI9aGaeqySde2aaWbaaS qabeaacqaH0oazaaGccaGGSaaaaa@385D@ and

γ δ = { i s δ 1 k i δ p i δ ( 1 p i δ ) h i δ ( h i δ ) } 1 i s δ 1 1 p i δ p ^ i 1 δ 1 h i δ y i t π i . ( 2.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHZoWzdaahaaWcbeqaaiabes7aKb aakiaai2dadaGadaqaamaaqafabeWcbaGaamyAaiabgIGiolaadoha daWgaaadbaGaeqiTdqMaeyOeI0IaaGymaaqabaaaleqaniabggHiLd GccaWGRbWaa0baaSqaaiaadMgaaeaacqaH0oazaaGccaWGWbWaa0ba aSqaaiaadMgaaeaacqaH0oazaaGcdaqadaqaaiaaigdacqGHsislca WGWbWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaawIcacaGLPaaa caWGObWaa0baaSqaaiaadMgaaeaacqaH0oazaaGcdaqadaqaaiaadI gadaqhaaWcbaGaamyAaaqaaiabes7aKbaaaOGaayjkaiaawMcaamaa CaaaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiq qacqWFKksLaaaakiaawUhacaGL9baadaahaaWcbeqaaiabgkHiTiaa igdaaaGcdaaeqbqabSqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqaai abes7aKjabgkHiTiaaigdaaeqaaaWcbeqdcqGHris5aOGaaGjbVpaa laaabaGaaGymaiabgkHiTiaadchadaqhaaWcbaGaamyAaaqaaiabes 7aKbaaaOqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaa ykW7cqGHsgIRcaaMc8UaeqiTdqMaaGPaVlabgkHiTiaaykW7caaIXa aaaaaakiaaysW7caWGObWaa0baaSqaaiaadMgaaeaacqaH0oazaaGc caaMe8+aaSaaaeaacaWG5bWaaSbaaSqaaiaadMgacaWG0baabeaaaO qaaiabec8aWnaaBaaaleaacaWGPbaabeaaaaGccaaIUaGaaGzbVlaa ywW7caaMf8UaaiikaiaaikdacaGGUaGaaGOnaiaacMcaaaa@9A75@

From (2.5), we obtain that

E { Y ^ lin , t ( t ) | s t 1 } = Y ^ t 1 ( t ) , ( 2.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGfbWaaiWaaeaaceWGzbGbaKaada WgaaWcbaGaaeiBaiaabMgacaqGUbGaaGzaVlaaiYcacaaMe8UaamiD aaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaadaabbaqaaiaayk W7aiaawEa7aiaadohadaWgaaWcbaGaamiDaiabgkHiTiaaigdaaeqa aaGccaGL7bGaayzFaaGaaGypaiqadMfagaqcamaaBaaaleaacaWG0b GaeyOeI0IaaGymaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaa caaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlai aaiEdacaGGPaaaaa@5870@

with Y ^ t 1 ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDai abgkHiTiaaigdaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaaa aa@3813@ the estimator of Y ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGzbWaaeWaaeaacaWG0baacaGLOa Gaayzkaaaaaa@352C@ computed on s t 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaadshacqGHsi slcaaIXaaabeaakiaac6caaaa@364D@ Using a proof by induction, it follows from (2.4) and (2.7) that Y ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@33DF@ is approximately unbiased for Y ( t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGzbWaaeWaaeaacaWG0baacaGLOa GaayzkaaGaaiOlaaaa@35DE@ Also, the variance of Y ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@33DF@ may be asymptotically approximated by

V app ( Y ^ t ) = V ( i s 0 y i t π i ) + E [ δ = 1 t V { Y ^ lin , δ ( t ) | s δ 1 } ] . ( 2.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaSbaaSqaaiaabggacaqGWb GaaeiCaaqabaGcdaqadaqaaiqadMfagaqcamaaBaaaleaacaWG0baa beaaaOGaayjkaiaawMcaaiaai2dacaWGwbWaaeWaaeaadaaeqbqabS qaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqaaiaaicdaaeqaaaWcbeqd cqGHris5aOWaaSaaaeaacaWG5bWaaSbaaSqaaiaadMgacaWG0baabe aaaOqaaiabec8aWnaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGL PaaacqGHRaWkcaWGfbWaamWaaeaadaaeWbqabSqaaiabes7aKjaai2 dacaaIXaaabaGaamiDaaqdcqGHris5aOGaaGjbVlaadAfadaGadaqa aiqadMfagaqcamaaBaaaleaacaqGSbGaaeyAaiaab6gacaaMb8UaaG ilaiaaykW7cqaH0oazaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaWaaqqaaeaacaaMc8oacaGLhWoacaWGZbWaaSbaaSqaaiabes7aKj abgkHiTiaaigdaaeqaaaGccaGL7bGaayzFaaaacaGLBbGaayzxaaGa aGOlaiaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6caca aI4aGaaiykaaaa@75A6@

The first term in the right-hand side of (2.8) is the variance due to the sampling design, that we note as V p ( Y ^ t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGWbaaaO WaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGaamiDaaqabaaakiaawIca caGLPaaacaGGUaaaaa@382B@ The second term in the right-hand side of (2.8) is the variance due to non-response, that we note as V nr ( Y ^ t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaqGUbGaae OCaaaakmaabmaabaGabmywayaajaWaaSbaaSqaaiaadshaaeqaaaGc caGLOaGaayzkaaGaaiOlaaaa@391C@ From (2.5), this asymptotic variance is given by

V nr ( Y ^ t ) = E ( δ = 1 t V nr δ ( Y ^ t ) ) , ( 2.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaqGUbGaae OCaaaakmaabmaabaGabmywayaajaWaaSbaaSqaaiaadshaaeqaaaGc caGLOaGaayzkaaGaaGypaiaadweadaqadaqaamaaqahabeWcbaGaeq iTdqMaaGypaiaaigdaaeaacaWG0baaniabggHiLdGccaaMe8UaamOv amaaCaaaleqabaGaaeOBaiaabkhacqaH0oazaaGcdaqadaqaaiqadM fagaqcamaaBaaaleaacaWG0baabeaaaOGaayjkaiaawMcaaaGaayjk aiaawMcaaiaaiYcacaaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaik dacaGGUaGaaGyoaiaacMcaaaa@5638@

where

V nr δ ( Y ^ t ) = i s δ 1 p i δ ( 1 p i δ ) ( y i t π i p ^ i 1 δ 1 p i δ k i δ ( h i δ ) γ δ ) 2 . ( 2.10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaqGUbGaae OCaiabes7aKbaakmaabmaabaGabmywayaajaWaaSbaaSqaaiaadsha aeqaaaGccaGLOaGaayzkaaGaaGypamaaqafabeWcbaGaamyAaiabgI GiolaadohadaWgaaadbaGaeqiTdqMaeyOeI0IaaGymaaqabaaaleqa niabggHiLdGccaWGWbWaa0baaSqaaiaadMgaaeaacqaH0oazaaGcda qadaqaaiaaigdacqGHsislcaWGWbWaa0baaSqaaiaadMgaaeaacqaH 0oazaaaakiaawIcacaGLPaaacaaMe8+aaeWaaeaadaWcaaqaaiaadM hadaWgaaWcbaGaamyAaiaadshaaeqaaaGcbaGaeqiWda3aaSbaaSqa aiaadMgaaeqaaOGabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXa GaaGPaVlabgkziUkaaykW7cqaH0oazcaaMc8UaeyOeI0IaaGPaVlaa igdaaaGccaWGWbWaa0baaSqaaiaadMgaaeaacqaH0oazaaaaaOGaey OeI0Iaam4AamaaDaaaleaacaWGPbaabaGaeqiTdqgaaOWaaeWaaeaa caWGObWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaawIcacaGLPa aadaahaaWcbeqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KB LbaceeGae8hPIujaaOGaeq4SdC2aaWbaaSqabeaacqaH0oazaaaaki aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaaMb8UaaGOlaiaa ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaIXaGaaG imaiaacMcaaaa@9081@

We note that for each of its component δ = 1, , t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0bGaaiilaaaa@3C44@ the term V nr δ ( Y ^ t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaqGUbGaae OCaiabes7aKbaakmaabmaabaGabmywayaajaWaaSbaaSqaaiaadsha aeqaaaGccaGLOaGaayzkaaaaaa@3A0F@ in (2.10) includes a centering term k i δ ( h i δ ) γ δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGcdaqadaqaaiaadIgadaqhaaWcbaGaamyAaaqaaiabes7a KbaaaOGaayjkaiaawMcaamaaCaaaleqabaWefv3ySLgznfgDOfdary qr1ngBPrginfgDObYtUvgaiqqacqWFKksLaaGccqaHZoWzdaahaaWc beqaaiabes7aKbaakiaaygW7caGGSaaaaa@4C24@ which is essentially a prediction of ( π i p ^ i 1 δ 1 p i δ ) 1 y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiabec8aWnaaBaaaleaaca WGPbaabeaakiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaa ykW7cqGHsgIRcaaMc8UaeqiTdqMaaGPaVlabgkHiTiaaykW7caaIXa aaaOGaamiCamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGccaGLOaGa ayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamyEamaaBaaale aacaWGPbaabeaaaaa@4C37@ by means of regressors h i δ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGObWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaMb8UaaiOlaaaa@37BF@ This centering is due to the estimation of the response probabilities. Suppressing these centering terms, equations (2.9) and (2.10) would lead to the variance of the estimator of Y ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGzbWaaeWaaeaacaWG0baacaGLOa Gaayzkaaaaaa@352C@ we would obtain by replacing in (2.3) the estimated probabilities by their true values. The variance of this estimator is usually larger than that of the reweighted estimator in (2.3); see also Beaumont (2005), equation (5.7) and Kim and Kim (2007), equation (17), for the case t = 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdacaGGUaaaaa@34F9@

2.4  Variance estimation

At time t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiilaaaa@3375@ an approximately unbiased estimator for the variance due to the sampling design V p ( Y ^ t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGWbaaaO WaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGaamiDaaqabaaakiaawIca caGLPaaaaaa@3779@ is

V ^ t p ( Y ^ t ) = i , j s t Δ i j π i j 1 p ^ i j 1 t y i t π i y j t π j , ( 2.11 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaadchaaaGcdaqadaqaaiqadMfagaqcamaaBaaaleaacaWG0baa beaaaOGaayjkaiaawMcaaiaai2dadaaeqbqabSqaaiaadMgacaaISa GaamOAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaaaleqaniab ggHiLdGccaaMe8+aaSaaaeaacqqHuoardaWgaaWcbaGaamyAaiaadQ gaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadMgacaWGQbaabeaaaaGc caaMe8+aaSaaaeaacaaIXaaabaGabmiCayaajaWaa0baaSqaaiaadM gacaWGQbaabaGaaGymaiaaykW7cqGHsgIRcaaMc8UaamiDaaaaaaGc caaMe8+aaSaaaeaacaWG5bWaaSbaaSqaaiaadMgacaWG0baabeaaaO qaaiabec8aWnaaBaaaleaacaWGPbaabeaaaaGccaaMe8+aaSaaaeaa caWG5bWaaSbaaSqaaiaadQgacaWG0baabeaaaOqaaiabec8aWnaaBa aaleaacaWGQbaabeaaaaGccaaISaGaaGzbVlaaywW7caaMf8UaaGzb VlaacIcacaaIYaGaaiOlaiaaigdacaaIXaGaaiykaaaa@7123@

where p ^ i j 1 t δ = 1 t p ^ i j δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaamyAai aadQgaaeaacaaIXaGaaGPaVlabgkziUkaaykW7caWG0baaaOGaeyyy IO7aaebmaeqaleaacqaH0oazcaaI9aGaaGymaaqaaiaadshaa0Gaey 4dIunakiaaysW7ceWGWbGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaa cqaH0oazaaGccaaMb8Uaaiilaaaa@4C06@ and where p ^ i j δ = p ^ i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaamyAai aadQgaaeaacqaH0oazaaGccaaI9aGabmiCayaajaWaa0baaSqaaiaa dMgaaeaacqaH0oazaaaaaa@3B16@ if i = j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaadQgacaGGSaaaaa@3520@ and p ^ i j δ = p ^ i δ p ^ j δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaamyAai aadQgaaeaacqaH0oazaaGccaaI9aGabmiCayaajaWaa0baaSqaaiaa dMgaaeaacqaH0oazaaGcceWGWbGbaKaadaqhaaWcbaGaamOAaaqaai abes7aKbaaaaa@3EE6@ otherwise. Following equation (25) in Kim and Kim (2007), V nr ( Y ^ t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGwbWaaWbaaSqabeaacaqGUbGaae OCaaaakmaabmaabaGabmywayaajaWaaSbaaSqaaiaadshaaeqaaaGc caGLOaGaayzkaaaaaa@386A@ may be approximately unbiasedly estimated at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0baaaa@32C5@ by

V ^ t nr ( Y ^ t ) = δ = 1 t V ^ t nr δ ( Y ^ t ) ( 2.12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaab6gacaqGYbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGa amiDaaqabaaakiaawIcacaGLPaaacaaI9aWaaabCaeqaleaacqaH0o azcaaI9aGaaGymaaqaaiaadshaa0GaeyyeIuoakiaaysW7ceWGwbGb aKaadaqhaaWcbaGaamiDaaqaaiaab6gacaqGYbGaeqiTdqgaaOWaae WaaeaaceWGzbGbaKaadaWgaaWcbaGaamiDaaqabaaakiaawIcacaGL PaaacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaai OlaiaaigdacaaIYaGaaiykaaaa@5783@

where

V ^ t nr δ ( Y ^ t ) = i s t p ^ i δ ( 1 p ^ i δ ) p ^ i δ t ( y i t π i p ^ i 1 δ k i δ ( h ^ i δ ) γ ^ t δ ) 2 , ( 2.13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaab6gacaqGYbGaeqiTdqgaaOWaaeWaaeaaceWGzbGbaKaadaWg aaWcbaGaamiDaaqabaaakiaawIcacaGLPaaacaaI9aWaaabuaeqale aacaWGPbGaeyicI4Saam4CamaaBaaameaacaWG0baabeaaaSqab0Ga eyyeIuoakmaalaaabaGabmiCayaajaWaa0baaSqaaiaadMgaaeaacq aH0oazaaGcdaqadaqaaiaaigdacqGHsislceWGWbGbaKaadaqhaaWc baGaamyAaaqaaiabes7aKbaaaOGaayjkaiaawMcaaaqaaiqadchaga qcamaaDaaaleaacaWGPbaabaGaeqiTdqMaaGPaVlabgkziUkaaykW7 caWG0baaaaaakmaabmaabaWaaSaaaeaacaWG5bWaaSbaaSqaaiaadM gacaWG0baabeaaaOqaaiabec8aWnaaBaaaleaacaWGPbaabeaakiqa dchagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaaykW7cqGHsgIRca aMc8UaeqiTdqgaaaaakiabgkHiTiaadUgadaqhaaWcbaGaamyAaaqa aiabes7aKbaakmaabmaabaGabmiAayaajaWaa0baaSqaaiaadMgaae aacqaH0oazaaaakiaawIcacaGLPaaadaahaaWcbeqaamrr1ngBPrwt HrhAXaqeguuDJXwAKbstHrhAG8KBLbaceeGae8hPIujaaOGafq4SdC MbaKaadaqhaaWcbaGaamiDaaqaaiabes7aKbaaaOGaayjkaiaawMca amaaCaaaleqabaGaaGOmaaaakiaaygW7caaISaGaaGzbVlaaywW7ca aMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaigdacaaIZaGaaiykaaaa @9041@

h ^ i δ = h ( z i , α ^ δ ) , ( 2.14 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGObGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaakiaai2dacaWGObWaaeWaaeaacaWG6bWaaSbaaSqa aiaadMgaaeqaaOGaaGzaVlaaiYcacaaMe8UafqySdeMbaKaadaahaa Wcbeqaaiabes7aKbaaaOGaayjkaiaawMcaaiaaiYcacaaMf8UaaGzb VlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaGymaiaaisdacaGGPa aaaa@4D78@

γ ^ t δ = { i s t k i δ p ^ i δ ( 1 p ^ i δ ) p ^ i δt h ^ i δ ( h ^ i δ ) } 1 i s t 1 p ^ i δ p ^ i 1t h ^ i δ y it π i .(2.15) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b aabaGaeqiTdqgaaOGaaGypamaacmaabaWaaabuaeqaleaacaWGPbGa eyicI4Saam4CamaaBaaameaacaWG0baabeaaaSqab0GaeyyeIuoaki aaysW7caWGRbWaa0baaSqaaiaadMgaaeaacqaH0oazaaGcdaWcaaqa aiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaOWaaeWaae aacaaIXaGaeyOeI0IabmiCayaajaWaa0baaSqaaiaadMgaaeaacqaH 0oazaaaakiaawIcacaGLPaaaaeaaceWGWbGbaKaadaqhaaWcbaGaam yAaaqaaiabes7aKjaaykW7cqGHsgIRcaaMc8UaamiDaaaaaaGcceWG ObGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaakmaabmaabaGabm iAayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaawIcacaGL PaaadaahaaWcbeqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8 KBLbaceeGae8hPIujaaaGccaGL7bGaayzFaaWaaWbaaSqabeaacqGH sislcaaIXaaaaOWaaabuaeqaleaacaWGPbGaeyicI4Saam4CamaaBa aameaacaWG0baabeaaaSqab0GaeyyeIuoakmaalaaabaGaaGymaiab gkHiTiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGcba GabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkzi UkaaykW7caWG0baaaaaakiaaysW7caaMe8UabmiAayaajaWaa0baaS qaaiaadMgaaeaacqaH0oazaaGccaaMe8UaaGjbVpaalaaabaGaamyE amaaBaaaleaacaWGPbGaamiDaaqabaaakeaacqaHapaCdaWgaaWcba GaamyAaaqabaaaaOGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7caGG OaGaaGOmaiaac6cacaaIXaGaaGynaiaacMcaaaa@A109@

This leads to the global variance estimator at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0baaaa@32C5@

V ^ t ( Y ^ t ) = V ^ t p ( Y ^ t ) + V ^ t nr ( Y ^ t ) . ( 2.16 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaWgaaWcbaGaamiDaa qabaGcdaqadaqaaiqadMfagaqcamaaBaaaleaacaWG0baabeaaaOGa ayjkaiaawMcaaiaai2daceWGwbGbaKaadaqhaaWcbaGaamiDaaqaai aadchaaaGcdaqadaqaaiqadMfagaqcamaaBaaaleaacaWG0baabeaa aOGaayjkaiaawMcaaiabgUcaRiqadAfagaqcamaaDaaaleaacaWG0b aabaGaaeOBaiaabkhaaaGcdaqadaqaaiqadMfagaqcamaaBaaaleaa caWG0baabeaaaOGaayjkaiaawMcaaiaai6cacaaMf8UaaGzbVlaayw W7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaigdacaaI2aGaaiyk aaaa@5451@

A simplified estimator of the variance due to non-response is obtained by ignoring the prediction terms k i δ ( h ^ i δ ) γ ^ t δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGcdaqadaqaaiqadIgagaqcamaaDaaaleaacaWGPbaabaGa eqiTdqgaaaGccaGLOaGaayzkaaWaaWbaaSqabeaatuuDJXwAK1uy0H wmaeHbfv3ySLgzG0uy0Hgip5wzaGabbiab=rQivcaakiqbeo7aNzaa jaWaa0baaSqaaiaadshaaeaacqaH0oazaaaaaa@4AF9@ for each of the δ = 1, , t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0baaaa@3B94@ variance components. After some algebra, this leads to the simplified variance estimator

V ^ t , simp nr { Y ^ t ( t ) } = i s t 1 p ^ i 1 t ( p ^ i 1 t ) 2 ( y i t π i ) 2 . ( 2.17 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDai aaygW7caaISaGaaGjbVlaabohacaqGPbGaaeyBaiaabchaaeaacaqG UbGaaeOCaaaakmaacmaabaGabmywayaajaWaaSbaaSqaaiaadshaae qaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGL7bGaayzFaaGa aGypamaaqafabeWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaam iDaaqabaaaleqaniabggHiLdGccaaMe8+aaSaaaeaacaaIXaGaeyOe I0IabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgk ziUkaaykW7caWG0baaaaGcbaWaaeWaaeaaceWGWbGbaKaadaqhaaWc baGaamyAaaqaaiaaigdacaaMc8UaeyOKH4QaaGPaVlaadshaaaaaki aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaOWaaeWaaeaadaWc aaqaaiaadMhadaWgaaWcbaGaamyAaiaadshaaeqaaaGcbaGaeqiWda 3aaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaakiaaygW7caaIUaGaaGzbVlaaywW7caaMf8UaaGzbVl aaywW7caGGOaGaaGOmaiaac6cacaaIXaGaaG4naiaacMcaaaa@7A14@

The main advantage of this simplified variance estimator is that it only requires the knowledge of the estimated response probabilities. On the other hand, the computation of the variance estimator in (2.12) requires the knowledge of the response models used at all times. The simplified variance estimator is therefore of particular interest for secondary users of the survey data, for which the estimated response probabilities may be the only available information related to the response modeling. This simplified variance estimator will tend to overestimate the variance due to non-response of ( Y ^ t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiqadMfagaqcamaaBaaale aacaWG0baabeaaaOGaayjkaiaawMcaaaaa@3572@ if the prediction term k i δ ( h i δ ) γ δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGcdaqadaqaaiaadIgadaqhaaWcbaGaamyAaaqaaiabes7a KbaaaOGaayjkaiaawMcaamaaCaaaleqabaWefv3ySLgznfgDOfdary qr1ngBPrginfgDObYtUvgaiqqacqWFKksLaaGccqaHZoWzdaahaaWc beqaaiabes7aKbaaaaa@49E0@ partly explains ( π i p ^ i 1 δ 1 p i δ ) 1 y i t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiabec8aWnaaBaaaleaaca WGPbaabeaakiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaa ykW7cqGHsgIRcaaMc8UaeqiTdqMaaGPaVlabgkHiTiaaykW7caaIXa aaaOGaamiCamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGccaGLOaGa ayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamyEamaaBaaale aacaWGPbGaamiDaaqabaGccaaMb8UaaiOlaaaa@4F76@

2.5  Application to the logistic regression model

In the particular case when a logistic regression model is used at each time δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaGGSaaaaa@3421@ the model (2.1) may be rewritten as

logit ( p i δ ) = ( z i δ ) α δ . ( 2.18 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaqGSbGaae4BaiaabEgacaqGPbGaae iDamaabmaabaGaamiCamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGc caGLOaGaayzkaaGaaGypamaabmaabaGaamOEamaaDaaaleaacaWGPb aabaGaeqiTdqgaaaGccaGLOaGaayzkaaWaaWbaaSqabeaatuuDJXwA K1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGabbiab=rQivcaakiabeg 7aHnaaCaaaleqabaGaeqiTdqgaaOGaaGOlaiaaywW7caaMf8UaaGzb VlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaGymaiaaiIdacaGGPa aaaa@5DB8@

We obtain h ^ i δ = z i δ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGObGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaakiaai2dacaWG6bWaa0baaSqaaiaadMgaaeaacqaH 0oazaaGccaaMb8Uaaiilaaaa@3C5D@ and the estimator for the variance due to non-response is given by (2.12), with

V ^ t nr δ ( Y ^ t ) = i s t p ^ i δ ( 1 p ^ i δ ) p ^ i δ t ( y i t π i p ^ i 1 δ k i δ ( z i δ ) γ ^ t δ ) 2 , ( 2.19 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaab6gacaqGYbGaeqiTdqgaaOWaaeWaaeaaceWGzbGbaKaadaWg aaWcbaGaamiDaaqabaaakiaawIcacaGLPaaacaaI9aWaaabuaeqale aacaWGPbGaeyicI4Saam4CamaaBaaameaacaWG0baabeaaaSqab0Ga eyyeIuoakiaaysW7daWcaaqaaiqadchagaqcamaaDaaaleaacaWGPb aabaGaeqiTdqgaaOWaaeWaaeaacaaIXaGaeyOeI0IabmiCayaajaWa a0baaSqaaiaadMgaaeaacqaH0oazaaaakiaawIcacaGLPaaaaeaace WGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKjaaykW7cqGHsgIR caaMc8UaamiDaaaaaaGccaaMe8+aaeWaaeaadaWcaaqaaiaadMhada WgaaWcbaGaamyAaiaadshaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaa dMgaaeqaaOGabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaG PaVlabgkziUkaaykW7cqaH0oazaaaaaOGaeyOeI0Iaam4AamaaDaaa leaacaWGPbaabaGaeqiTdqgaaOWaaeWaaeaacaWG6bWaa0baaSqaai aadMgaaeaacqaH0oazaaaakiaawIcacaGLPaaadaahaaWcbeqaamrr 1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbaceeGae8hPIujaaO Gafq4SdCMbaKaadaqhaaWcbaGaamiDaaqaaiabes7aKbaaaOGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaakiaaygW7caaISaGaaGzbVl aaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaigdacaaI5aGa aiykaaaa@9363@

γ ^ t δ = { i s t k i δ p ^ i δ ( 1 p ^ i δ ) p ^ i δ t z i δ ( z i δ ) } 1 i s t 1 p ^ i δ p ^ i 1 t z i δ y i t π i . ( 2.20 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b aabaGaeqiTdqgaaOGaaGypamaacmaabaWaaabuaeqaleaacaWGPbGa eyicI4Saam4CamaaBaaameaacaWG0baabeaaaSqab0GaeyyeIuoaki aaysW7caWGRbWaa0baaSqaaiaadMgaaeaacqaH0oazaaGcdaWcaaqa aiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaOWaaeWaae aacaaIXaGaeyOeI0IabmiCayaajaWaa0baaSqaaiaadMgaaeaacqaH 0oazaaaakiaawIcacaGLPaaaaeaaceWGWbGbaKaadaqhaaWcbaGaam yAaaqaaiabes7aKjaaykW7cqGHsgIRcaaMc8UaamiDaaaaaaGccaWG 6bWaa0baaSqaaiaadMgaaeaacqaH0oazaaGcdaqadaqaaiaadQhada qhaaWcbaGaamyAaaqaaiabes7aKbaaaOGaayjkaiaawMcaamaaCaaa leqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiqqacq WFKksLaaaakiaawUhacaGL9baadaahaaWcbeqaaiabgkHiTiaaigda aaGcdaaeqbqabSqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqaaiaads haaeqaaaWcbeqdcqGHris5aOGaaGjbVpaalaaabaGaaGymaiabgkHi TiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGcbaGabm iCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkziUkaa ykW7caWG0baaaaaakiaaysW7caWG6bWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaMe8+aaSaaaeaacaWG5bWaaSbaaSqaaiaadMgacaWG 0baabeaaaOqaaiabec8aWnaaBaaaleaacaWGPbaabeaaaaGccaaIUa GaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaikda caaIWaGaaiykaaaa@9F7E@

If the reweighted estimator is computed at time t = 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdacaGGSaaaaa@34F7@ the estimator in (2.12) for the variance due to non-response may be rewritten as

V ^ 1 nr ( Y ^ 1 ) = i s 1 ( 1 p ^ i 1 ) ( y i 1 π i p ^ i 1 k i 1 ( z i 1 ) γ ^ 1 1 ) 2 . ( 2.21 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaaGymaa qaaiaab6gacaqGYbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGa aGymaaqabaaakiaawIcacaGLPaaacaaI9aWaaabuaeqaleaacaWGPb GaeyicI4Saam4CamaaBaaameaacaaIXaaabeaaaSqab0GaeyyeIuoa kiaaysW7daqadaqaaiaaigdacqGHsislceWGWbGbaKaadaqhaaWcba GaamyAaaqaaiaaigdaaaaakiaawIcacaGLPaaadaqadaqaamaalaaa baGaamyEamaaBaaaleaacaWGPbGaaGymaaqabaaakeaacqaHapaCda WgaaWcbaGaamyAaaqabaGcceWGWbGbaKaadaqhaaWcbaGaamyAaaqa aiaaigdaaaaaaOGaeyOeI0Iaam4AamaaDaaaleaacaWGPbaabaGaaG ymaaaakmaabmaabaGaamOEamaaDaaaleaacaWGPbaabaGaaGymaaaa aOGaayjkaiaawMcaamaaCaaaleqabaWefv3ySLgznfgDOfdaryqr1n gBPrginfgDObYtUvgaiqqacqWFKksLaaGccuaHZoWzgaqcamaaDaaa leaacaaIXaaabaGaaGymaaaaaOGaayjkaiaawMcaamaaCaaaleqaba GaaGOmaaaakiaai6cacaaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaa ikdacaGGUaGaaGOmaiaaigdacaGGPaaaaa@7572@

If the reweighted estimator is computed at time t = 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaikdacaGGSaaaaa@34F8@ the estimator in (2.12) for the variance due to non-response may be rewritten as

V ^ 2 nr ( Y ^ 2 ) = i s 2 ( 1 p ^ i 1 ) p ^ i 2 ( y i2 π i p ^ i 1 k i 1 ( z i 1 ) γ ^ 2 1 ) 2 + i s 2 ( 1 p ^ i 2 ) ( y i2 π i p ^ i 1 p ^ i 2 k i 2 ( z i 2 ) γ ^ 2 2 ) 2 .(2.22) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaafaqaaeGacaaabaGabmOvayaajaWaa0 baaSqaaiaaikdaaeaacaqGUbGaaeOCaaaakmaabmaabaGabmywayaa jaWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaabaGaaGypam aaqafabeWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaaGOmaaqa baaaleqaniabggHiLdGcdaWcaaqaamaabmaabaGaaGymaiabgkHiTi qadchagaqcamaaDaaaleaacaWGPbaabaGaaGymaaaaaOGaayjkaiaa wMcaaaqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGOmaaaaaa GcdaqadaqaamaalaaabaGaamyEamaaBaaaleaacaWGPbGaaGOmaaqa baaakeaacqaHapaCdaWgaaWcbaGaamyAaaqabaGcceWGWbGbaKaada qhaaWcbaGaamyAaaqaaiaaigdaaaaaaOGaeyOeI0Iaam4AamaaDaaa leaacaWGPbaabaGaaGymaaaakmaabmaabaGaamOEamaaDaaaleaaca WGPbaabaGaaGymaaaaaOGaayjkaiaawMcaamaaCaaaleqabaWefv3y SLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiqqacqWFKksLaaGccu aHZoWzgaqcamaaDaaaleaacaaIYaaabaGaaGymaaaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaaaOqaaaqaaiabgUcaRmaaqafabe WcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaaGOmaaqabaaaleqa niabggHiLdGcdaqadaqaaiaaigdacqGHsislceWGWbGbaKaadaqhaa WcbaGaamyAaaqaaiaaikdaaaaakiaawIcacaGLPaaadaqadaqaamaa laaabaGaamyEamaaBaaaleaacaWGPbGaaGOmaaqabaaakeaacqaHap aCdaWgaaWcbaGaamyAaaqabaGcceWGWbGbaKaadaqhaaWcbaGaamyA aaqaaiaaigdaaaGcceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiaaik daaaaaaOGaeyOeI0Iaam4AamaaDaaaleaacaWGPbaabaGaaGOmaaaa kmaabmaabaGaamOEamaaDaaaleaacaWGPbaabaGaaGOmaaaaaOGaay jkaiaawMcaamaaCaaaleqabaGae8hPIujaaOGafq4SdCMbaKaadaqh aaWcbaGaaGOmaaqaaiaaikdaaaaakiaawIcacaGLPaaadaahaaWcbe qaaiaaikdaaaGccaaMb8UaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7 caaMf8UaaiikaiaaikdacaGGUaGaaGOmaiaaikdacaGGPaaaaaaa@A347@

In practice, the model of Response Homogeneity Groups (RHG) is often assumed when correcting for unit non-response. Under this model, it is assumed that at each time δ = 1, , t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0bGaaiilaaaa@3C44@ the sub-sample s δ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiabes7aKjabgk HiTiaaigdaaeqaaaaa@363D@ may be partitioned into C ( δ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGdbWaaeWaaeaacqaH0oazcqGHsi slcaaIXaaacaGLOaGaayzkaaaaaa@376A@ groups s δ 1 c , c = 1, , C ( δ 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaa0baaSqaaiabes7aKjabgk HiTiaaigdaaeaacaWGJbaaaOGaaGzaVlaaiYcacaaMe8Uaam4yaiaa i2dacaaIXaGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlaadoeada qadaqaaiabes7aKjabgkHiTiaaigdaaiaawIcacaGLPaaacaGGSaaa aa@495D@ such that the response probability p i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@3581@ is constant inside a group. This model is a particular case of the logistic regression model in (2.18), obtained with

z i δ = [ 1 { i s δ 1 1 } , , 1 { i s δ 1 C ( δ 1 ) } ] , ( 2.23 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG6bWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaI9aWaamWaaeaacaaIXaWaaiWaaeaacaaMc8UaamyA aiabgIGiolaadohadaqhaaWcbaGaeqiTdqMaeyOeI0IaaGymaaqaai aaigdaaaaakiaawUhacaGL9baacaaISaGaaGjbVlablAciljaaiYca caaMe8UaaGymamaacmaabaGaaGPaVlaadMgacqGHiiIZcaWGZbWaa0 baaSqaaiabes7aKjabgkHiTiaaigdaaeaacaWGdbWaaeWaaeaacqaH 0oazcqGHsislcaaIXaaacaGLOaGaayzkaaaaaaGccaGL7bGaayzFaa aacaGLBbGaayzxaaWaaWbaaSqabeaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGabbiab=rQivcaakiaaygW7caaISaGaaGzbVl aaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaIYaGa aG4maiaacMcaaaa@74FF@

and the variance due to non-response is estimated accordingly. Explicit formulas are given in Appendix.


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