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    5. Conclusion

    John Preston

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    This paper extends a number of the modified regression estimators to business surveys with survey frames that change over time, due to the addition of "births� and the deletion of "deaths�. The results of the simulation study indicate that the magnitude of the bias of these various modified regression estimators is negligible. The "best� estimator was the compromise modified regression estimator which led to significant efficiency gains in both the point-in-time and movement estimates, with an appropriate choice of α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiabeg7aHb aa@3A9C@ eliminating the likelihood of the "drift� problem.

    Acknowledgements

    The views expressed in this paper are those of the author and do not necessarily reflect the views of the Australian Bureau of Statistics (ABS). The author would like to thank the anonymous referees and the associate editor for their valuable comments, and Dr Robert Clark at University of Woollongong for his constructive suggestions on an earlier draft of this manuscript.

    Appendix

    The expected values of the HT estimator for the "pseudo-composite auxiliary variables� Z ^ h * ( t ) = ∑ h = 1 H ∑ i ∈ s h * ( t ) w i * ( t ) z ( MR 2 ) i * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqahQfaga qcamaaDaaaleaacaWGObaabaGaaiOkamaabmqabaGaamiDaaGaayjk aiaawMcaaaaakiabg2da9maaqadabaWaaabeaeaacaWG3bWaa0baaS qaaiaadMgaaeaacaGGQaWaaeWabeaacaWG0baacaGLOaGaayzkaaaa aOGaaCOEamaaDaaaleaadaqadeqaaiaab2eacaqGsbGaaGOmaaGaay jkaiaawMcaaiaadMgaaeaacaGGQaWaaeWabeaacaWG0baacaGLOaGa ayzkaaaaaaqaaiaadMgacqGHiiIZcaWGZbWaa0baaWqaaiaadIgaae aacaGGQaWaaeWabeaacaWG0baacaGLOaGaayzkaaaaaaWcbeqdcqGH ris5aaWcbaGaamiAaiabg2da9iaaigdaaeaacaWGibaaniabggHiLd aaaa@5CED@ at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadshaaa a@39F6@ are given by:

    E [ Z ^ HT * ( t ) ] = E [ ∑ h = 1 H ∑ i ∈ s h * ( t ) w i * ( t ) z ( MR2 ) i * ( t ) ] = ∑ h = 1 H E [ ∑ i ∈ s h * ( t ) w i * ( t ) z ( MR2 ) i * ( t ) ] = ∑ h = 1 H N h ( t − 1 ) E [ ( ∑ i ∈ s h * ( t ) ∩ s h * ( t − 1 ) w i * ( t ) y i * ( t − 1 ) / ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) ) ] − ∑ h = 1 H N h ( t − 1 ) E [ ( ∑ i ∈ s h * ( t ) ∩ s h * ( t − 1 ) w i * ( t ) y i * ( t ) / ∑ i ∈ s h ( t ) ∩ s h * ( t − 1 ) w i * ( t ) ) ] + ∑ h = 1 H N h ( t − 1 ) ( ∑ i ∈ s h ( t ) ∩ s h * ( t − 1 ) w i * ( t ) / ∑ i ∈ s h ( t ) w i ( t ) ) E [ ∑ i ∈ s h * ( t ) ∩ s h * ( t − 1 ) w i * ( t ) y i * ( t ) / ∑ i ∈ s h ( t ) ∩ s h * ( t − 1 ) w i * ( t ) ] + ∑ h = 1 H N h ( t − 1 ) ( ∑ i ∈ s h ( t ) \ s h * ( t − 1 ) w i * ( t ) / ∑ i ∈ s h ( t ) w i ( t ) ) E [ ∑ i ∈ s h * ( t ) \ s h * ( t − 1 ) w i * ( t ) y i * ( t ) / ∑ i ∈ s h ( t ) \ s h * ( t − 1 ) w i * ( t ) ] = ∑ h = 1 H N h ( t − 1 ) Y ¯ h ( t − 1 ) − ∑ h = 1 H N h ( t − 1 ) Y ¯ h ( t ) + ∑ h = 1 H N h ( t − 1 ) Y ¯ h ( t ) = ∑ h = 1 H N h ( t − 1 ) Y ¯ h ( t − 1 ) = ∑ h = 1 H Y h ( t − 1 ) = Y ( t − 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaauaabaqGkm aaaaaabaGaamyramaadmaabaGabCOwayaajaWaa0baaSqaaiaabIea caqGubaabaGaaiOkamaabmaabaGaamiDaaGaayjkaiaawMcaaaaaaO Gaay5waiaaw2faaaqaaiabg2da9aqaaiaadweadaWadaqaamaaqada baWaaabeaeaacaWG3bWaa0baaSqaaiaadMgaaeaacaGGQaWaaeWaae aacaWG0baacaGLOaGaayzkaaaaaaqaaiaadMgacqGHiiIZcaWGZbWa a0baaWqaaiaadIgaaeaacaGGQaWaaeWaaeaacaWG0baacaGLOaGaay zkaaaaaaWcbeqdcqGHris5aOGaaCOEamaaDaaaleaadaqadaqaaiaa b2eacaqGsbGaaeOmaaGaayjkaiaawMcaaiaadMgaaeaacaGGQaWaae WaaeaacaWG0baacaGLOaGaayzkaaaaaaqaaiaadIgacqGH9aqpcaaI XaaabaGaamisaaqdcqGHris5aaGccaGLBbGaayzxaaaabaaabaGaey ypa0dabaWaaabmaeaacaWGfbWaamWaaeaadaaeqaqaaiaadEhadaqh aaWcbaGaamyAaaqaaiaacQcadaqadaqaaiaadshaaiaawIcacaGLPa 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GHPiYXcaWGZbWaa0baaWqaaiaadIgaaeaadaqadaqaaiaadshacqGH sislcaaIXaaacaGLOaGaayzkaaaaaaWcbeqdcqGHris5aaaaaOGaay jkaiaawMcaaaGaay5waiaaw2faaaWcbaGaamiAaiabg2da9iaaigda aeaacaWGibaaniabggHiLdaakeaaaeaacqGHsislaeaadaaeWaqaai aad6eadaqhaaWcbaGaamiAaaqaamaabmaabaGaamiDaiabgkHiTiaa igdaaiaawIcacaGLPaaaaaGccaWGfbWaamWaaeaadaqadaqaamaaly aabaWaaabeaeaacaWG3bWaa0baaSqaaiaadMgaaeaacaGGQaWaaeWa aeaacaWG0baacaGLOaGaayzkaaaaaaqaaiaadMgacqGHiiIZcaWGZb Waa0baaWqaaiaadIgaaeaacaGGQaWaaeWaaeaacaWG0baacaGLOaGa ayzkaaaaaSGaeyykICSaam4CamaaDaaameaacaWGObaabaGaaiOkam aabmaabaGaamiDaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaaaleqa niabggHiLdGccaWH5bWaa0baaSqaaiaadMgaaeaacaGGQaWaaeWaae aacaWG0baacaGLOaGaayzkaaaaaaGcbaWaaabeaeaacaWG3bWaa0ba aSqaaiaadMgaaeaacaGGQaWaaeWaaeaacaWG0baacaGLOaGaayzkaa aaaaqaaiaadMgacqGHiiIZcaWGZbWaa0baaWqaaiaadIgaaeaadaqa daqaaiaadshaaiaawIcacaGLPaaaaaWccqGHPiYXcaWGZbWaa0baaW qaaiaadIgaaeaacaGGQaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGa ayjkaiaawMcaaaaaaSqab0GaeyyeIuoaaaaakiaawIcacaGLPaaaai 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aakiaawIcacaGLPaaacaWGfbWaamWaaeaadaWcgaqaamaaqababaGa am4DamaaDaaaleaacaWGPbaabaGaaiOkamaabmaabaGaamiDaaGaay jkaiaawMcaaaaaaeaacaWGPbGaeyicI4Saam4CamaaDaaameaacaWG ObaabaGaaiOkamaabmaabaGaamiDaaGaayjkaiaawMcaaaaaliaacY facaWGZbWaa0baaWqaaiaadIgaaeaacaGGQaWaaeWaaeaacaWG0bGa eyOeI0IaaGymaaGaayjkaiaawMcaaaaaaSqab0GaeyyeIuoakiaahM hadaqhaaWcbaGaamyAaaqaaiaacQcadaqadaqaaiaadshaaiaawIca caGLPaaaaaaakeaadaaeqaqaaiaadEhadaqhaaWcbaGaamyAaaqaai aacQcadaqadaqaaiaadshaaiaawIcacaGLPaaaaaaabaGaamyAaiab gIGiolaadohadaqhaaadbaGaamiAaaqaamaabmaabaGaamiDaaGaay jkaiaawMcaaaaaliaacYfacaWGZbWaa0baaWqaaiaadIgaaeaacaGG QaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaaaS qab0GaeyyeIuoaaaaakiaawUfacaGLDbaaaSqaaiaadIgacqGH9aqp caaIXaaabaGaamisaaqdcqGHris5aaGcbaaabaGaeyypa0dabaWaaa bmaeaacaWGobWaa0baaSqaaiaadIgaaeaadaqadaqaaiaadshacqGH sislcaaIXaaacaGLOaGaayzkaaaaaOGabCywayaaraWaa0baaSqaai aadIgaaeaadaqadaqaaiaadshacqGHsislcaaIXaaacaGLOaGaayzk aaaaaaqaaiaadIgacqGH9aqpcaaIXaaabaGaamisaaqdcqGHris5aO GaeyOeI0YaaabmaeaacaWGobWaa0baaSqaaiaadIgaaeaadaqadaqa aiaadshacqGHsislcaaIXaaacaGLOaGaayzkaaaaaOGabCywayaara Waa0baaSqaaiaadIgaaeaadaqadaqaaiaadshaaiaawIcacaGLPaaa aaaabaGaamiAaiabg2da9iaaigdaaeaacaWGibaaniabggHiLdGccq GHRaWkdaaeWaqaaiaad6eadaqhaaWcbaGaamiAaaqaamaabmaabaGa amiDaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGcceWHzbGbaebada qhaaWcbaGaamiAaaqaamaabmaabaGaamiDaaGaayjkaiaawMcaaaaa aeaacaWGObGaeyypa0JaaGymaaqaaiaadIeaa0GaeyyeIuoaaOqaaa qaaiabg2da9aqaamaaqadabaGaamOtamaaDaaaleaacaWGObaabaWa aeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaakiqahM fagaqeamaaDaaaleaacaWGObaabaWaaeWaaeaacaWG0bGaeyOeI0Ia aGymaaGaayjkaiaawMcaaaaaaeaacaWGObGaeyypa0JaaGymaaqaai aadIeaa0GaeyyeIuoaaOqaaaqaaiabg2da9aqaamaaqadabaGaaCyw amaaDaaaleaacaWGObaabaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaa GaayjkaiaawMcaaaaaaeaacaWGObGaeyypa0JaaGymaaqaaiaadIea a0GaeyyeIuoaaOqaaaqaaiabg2da9aqaaiaahMfadaahaaWcbeqaam aabmaabaGaamiDaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGccaGG Uaaaaaaa@3B6C@

    The expected values of the HT estimator for the "pseudo-composite auxiliary variables� Z ^ HT * ( t ) = ∑ h = 1 H ∑ i ∈ s h * ( t ) w i * ( t ) z ( MRR ) i * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqahQfaga qcamaaDaaaleaacaqGibGaaeivaaqaaiaacQcadaqadeqaaiaadsha aiaawIcacaGLPaaaaaGccqGH9aqpdaaeWaqaamaaqababaGaam4Dam aaDaaaleaacaWGPbaabaGaaiOkamaabmqabaGaamiDaaGaayjkaiaa wMcaaaaakiaahQhadaqhaaWcbaWaaeWabeaacaqGnbGaaeOuaiaabk faaiaawIcacaGLPaaacaWGPbaabaGaaiOkamaabmqabaGaamiDaaGa ayjkaiaawMcaaaaaaeaacaWGPbGaeyicI4Saam4CamaaDaaameaaca WGObaabaGaaiOkamaabmqabaGaamiDaaGaayjkaiaawMcaaaaaaSqa b0GaeyyeIuoaaSqaaiaadIgacqGH9aqpcaaIXaaabaGaamisaaqdcq GHris5aaaa@5DBB@ at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadshaaa a@39F6@ are given by:

    E [ Z ^ HT * ( t ) ] = E [ ∑ h = 1 H ∑ i ∈ s h * ( t ) w i * ( t ) z ( MRR ) i * ( t ) ] = ∑ h = 1 H E [ ∑ i ∈ s h * ( t ) w i * ( t ) z ( MRR ) i * ( t ) ] . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaauaabaqGci aaaeaacaWGfbWaamWaaeaaceWHAbGbaKaadaqhaaWcbaGaaeisaiaa bsfaaeaacaGGQaWaaeWabeaacaWG0baacaGLOaGaayzkaaaaaaGcca GLBbGaayzxaaaabaGaeyypa0JaamyramaadmaabaWaaabmaeaadaae qaqaaiaadEhadaqhaaWcbaGaamyAaaqaaiaacQcadaqadeqaaiaads haaiaawIcacaGLPaaaaaaabaGaamyAaiabgIGiolaadohadaqhaaad baGaamiAaaqaaiaacQcadaqadeqaaiaadshaaiaawIcacaGLPaaaaa aaleqaniabggHiLdGccaWH6bWaa0baaSqaamaabmqabaGaaeytaiaa bkfacaqGsbaacaGLOaGaayzkaaGaamyAaaqaaiaacQcadaqadeqaai aadshaaiaawIcacaGLPaaaaaaabaGaamiAaiabg2da9iaaigdaaeaa caWGibaaniabggHiLdaakiaawUfacaGLDbaaaeaaaeaacqGH9aqpda aeWaqaaiaadweadaWadaqaamaaqababaGaam4DamaaDaaaleaacaWG PbaabaGaaiOkamaabmqabaGaamiDaaGaayjkaiaawMcaaaaaaeaaca WGPbGaeyicI4Saam4CamaaDaaameaacaWGObaabaGaaiOkamaabmqa baGaamiDaaGaayjkaiaawMcaaaaaaSqab0GaeyyeIuoakiaahQhada qhaaWcbaWaaeWabeaacaqGnbGaaeOuaiaabkfaaiaawIcacaGLPaaa caWGPbaabaGaaiOkamaabmqabaGaamiDaaGaayjkaiaawMcaaaaaaO Gaay5waiaaw2faaiaac6caaSqaaiaadIgacqGH9aqpcaaIXaaabaGa amisaaqdcqGHris5aaaaaaa@85F9@

    For the case where there are no units in the "pseudo-sample� in stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadIgaaa a@39EA@ at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadshaaa a@39F6@ which were not included in the "pseudo-sample� in stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadIgaaa a@39EA@ at time t − 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadshacq GHsislcaaIXaaaaa@3B9E@ ( s h * ( t ) \ s h * ( t − 1 ) = ∅ ) : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaabmqaba Gaam4CamaaDaaaleaacaWGObaabaGaaiOkamaabmqabaGaamiDaaGa ayjkaiaawMcaaaaakiaacYfacaWGZbWaa0baaSqaaiaadIgaaeaaca GGQaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaa kiabg2da9iabgwGigdGaayjkaiaawMcaaiaacQdaaaa@4AE5@

    E [ ∑ i ∈ s h * ( t ) w i * ( t ) z ( MRR ) i * ( t ) ] = N h ( t − 1 ) E [ ( ∑ i ∈ s h * ( t ) ∩ s h * ( t − 1 ) w i * ( t ) y i * ( t − 1 ) / ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) ) ] = N h ( t − 1 ) Y ¯ h ( t − 1 ) = Y h ( t − 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaauaabaqGdi aaaeaacaWGfbWaamWaaeaadaaeqaqaaiaadEhadaqhaaWcbaGaamyA aaqaaiaacQcadaqadeqaaiaadshaaiaawIcacaGLPaaaaaaabaGaam yAaiabgIGiolaadohadaqhaaadbaGaamiAaaqaaiaacQcadaqadeqa aiaadshaaiaawIcacaGLPaaaaaaaleqaniabggHiLdGccaWH6bWaa0 baaSqaamaabmqabaGaaeytaiaabkfacaqGsbaacaGLOaGaayzkaaGa amyAaaqaaiaacQcadaqadeqaaiaadshaaiaawIcacaGLPaaaaaaaki aawUfacaGLDbaaaeaacqGH9aqpcaWGobWaa0baaSqaaiaadIgaaeaa daqadaqaaiaadshacqGHsislcaaIXaaacaGLOaGaayzkaaaaaOGaam yramaadmaabaWaaeWaceaadaWcgaqaamaaqababaGaam4DamaaDaaa leaacaWGPbaabaGaaiOkamaabmqabaGaamiDaaGaayjkaiaawMcaaa aaaeaacaWGPbGaeyicI4Saam4CamaaDaaameaacaWGObaabaGaaiOk amaabmqabaGaamiDaaGaayjkaiaawMcaaaaaliabgMIihlaadohada qhaaadbaGaamiAaaqaaiaacQcadaqadaqaaiaadshacqGHsislcaaI XaaacaGLOaGaayzkaaaaaaWcbeqdcqGHris5aOGaaCyEamaaDaaale aacaWGPbaabaGaaiOkamaabmaabaGaamiDaiabgkHiTiaaigdaaiaa wIcacaGLPaaaaaaakeaadaaeqaqaaiaadEhadaqhaaWcbaGaamyAaa qaaiaacQcadaqadeqaaiaadshaaiaawIcacaGLPaaaaaaabaGaamyA aiabgIGiolaadohadaqhaaadbaGaamiAaaqaaiaacQcadaqadeqaai aadshaaiaawIcacaGLPaaaaaWccqGHPiYXcaWGZbWaa0baaWqaaiaa dIgaaeaadaqadaqaaiaadshacqGHsislcaaIXaaacaGLOaGaayzkaa aaaaWcbeqdcqGHris5aaaaaOGaayjkaiaawMcaaaGaay5waiaaw2fa aaqaaaqaaiabg2da9iaad6eadaqhaaWcbaGaamiAaaqaamaabmaaba GaamiDaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGcceWHzbGbaeba daqhaaWcbaGaamiAaaqaamaabmaabaGaamiDaiabgkHiTiaaigdaai aawIcacaGLPaaaaaaakeaaaeaacqGH9aqpcaWHzbWaa0baaSqaaiaa dIgaaeaadaqadaqaaiaadshacqGHsislcaaIXaaacaGLOaGaayzkaa aaaaaaaaa@AA00@

    and for the case where there are units in the "pseudo-sample� in stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadIgaaa a@39EA@ at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadshaaa a@39F6@ which were not included in the "pseudo-sample� in stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadIgaaa a@39EA@ at time t − 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadshacq GHsislcaaIXaaaaa@3B9E@ ( s h * ( t ) \ s h * ( t − 1 ) ≠ ∅ ) : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaabmqaba Gaam4CamaaDaaaleaacaWGObaabaGaaiOkamaabmqabaGaamiDaaGa ayjkaiaawMcaaaaakiaacYfacaWGZbWaa0baaSqaaiaadIgaaeaaca GGQaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaa kiabgcMi5kabgwGigdGaayjkaiaawMcaaiaacQdaaaa@4BA6@

    E [ ∑ i ∈ s h * ( t ) w i * ( t ) z ( MRR ) i * ( t ) ] = N h ( t − 1 ) ( ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) / ∑ i ∈ s h ( t ) w i ( t ) ) × E [ ∑ i ∈ s h * ( t ) ∩ s h * ( t − 1 ) w i * ( t ) y i * ( t − 1 ) / ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) ] + N h ( t − 1 ) ( ∑ i ∈ s h ( t ) \ s h * ( t − 1 ) w i * ( t ) / ∑ i ∈ s h ( t ) w i ( t ) ) × E [ ∑ i ∈ s h * ( t ) \ s h * ( t − 1 ) w i * ( t ) y i * ( t ) / ∑ i ∈ s h ( t ) \ s h * ( t − 1 ) w i * ( t ) ] − N h ( t − 1 ) ( ∑ i ∈ s h ( t ) \ s h * ( t − 1 ) w i * ( t ) / ∑ i ∈ s h ( t ) w i ( t ) ) × E [ ( ∑ i ∈ s h ( t ) ∩ s h * ( t − 1 ) w i * ( t ) y i * ( t ) / ∑ i ∈ s h ( t ) ∩ s h * ( t − 1 ) w i * ( t ) ) ] + N h ( t − 1 ) E [ ( ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) y i * ( t − 1 ) / ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) ) ] − N h ( t − 1 ) ( ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) / ∑ i ∈ s h ( t ) w i ( t ) ) × E [ ( ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) y i * ( t − 1 ) / ∑ i ∈ s h * ( t ) ∩ s h ( t − 1 ) w i * ( t ) ) ] = N h ( t − 1 ) Y ¯ h ( t − 1 ) = Y h ( t − 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaauaabaqGlm aaaaaabaGaamyramaadmaabaWaaabeaeaacaWG3bWaa0baaSqaaiaa dMgaaeaacaGGQaWaaeWabeaacaWG0baacaGLOaGaayzkaaaaaaqaai aadMgacqGHiiIZcaWGZbWaa0baaWqaaiaadIgaaeaacaGGQaWaaeWa beaacaWG0baacaGLOaGaayzkaaaaaaWcbeqdcqGHris5aOGaaCOEam aaDaaaleaadaqadaqaaiaab2eacaqGsbGaaeOuaaGaayjkaiaawMca aiaadMgaaeaacaGGQaWaaeWabeaacaWG0baacaGLOaGaayzkaaaaaa GccaGLBbGaayzxaaaabaGaeyypa0dabaGaamOtamaaDaaaleaacaWG ObaabaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaa aakmaabmaabaWaaSGbaeaadaaeqaqaaiaadEhadaqhaaWcbaGaamyA aaqaaiaacQcadaqadeqaaiaadshaaiaawIcacaGLPaaaaaaabaGaam yAaiabgIGiolaadohadaqhaaadbaGaamiAaaqaaiaacQcadaqadeqa 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ZbWaa0baaWqaaiaadIgaaeaadaqadaqaaiaadshaaiaawIcacaGLPa aaaaaaleqaniabggHiLdaaaaGccaGLOaGaayzkaaaabaaabaGaey41 aqlabaGaamyramaadmaabaWaaeWaceaadaWcgaqaamaaqababaGaam 4DamaaDaaaleaacaWGPbaabaGaaiOkamaabmqabaGaamiDaaGaayjk aiaawMcaaaaakiaahMhadaqhaaWcbaGaamyAaaqaaiaacQcadaqada qaaiaadshacqGHsislcaaIXaaacaGLOaGaayzkaaaaaaqaaiaadMga cqGHiiIZcaWGZbWaa0baaWqaaiaadIgaaeaacaGGQaWaaeWabeaaca WG0baacaGLOaGaayzkaaaaaSGaeyykICSaam4CamaaDaaameaacaWG ObaabaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaa aaaSqab0GaeyyeIuoaaOqaamaaqababaGaam4DamaaDaaaleaacaWG PbaabaGaaiOkamaabmqabaGaamiDaaGaayjkaiaawMcaaaaaaeaaca WGPbGaeyicI4Saam4CamaaDaaameaacaWGObaabaGaaiOkamaabmqa baGaamiDaaGaayjkaiaawMcaaaaaliabgMIihlaadohadaqhaaadba GaamiAaaqaamaabmaabaGaamiDaiabgkHiTiaaigdaaiaawIcacaGL PaaaaaaaleqaniabggHiLdaaaaGccaGLOaGaayzkaaaacaGLBbGaay zxaaaabaaabaGaeyypa0dabaGaamOtamaaDaaaleaacaWGObaabaWa aeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaakiqahM fagaqeamaaDaaaleaacaWGObaabaWaaeWaaeaacaWG0bGaeyOeI0Ia aGymaaGaayjkaiaawMcaaaaaaOqaaaqaaiabg2da9aqaaiaahMfada qhaaWcbaGaamiAaaqaamaabmaabaGaamiDaiabgkHiTiaaigdaaiaa wIcacaGLPaaaaaaaaaaa@41E6@

    hence E [ Z ^ HT * ( t ) ] = ∑ h = 1 H Y h ( t − 1 ) = Y ( t − 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadweada WadaqaaiqahQfagaqcamaaDaaaleaacaqGibGaaeivaaqaaiaacQca daqadeqaaiaadshaaiaawIcacaGLPaaaaaaakiaawUfacaGLDbaacq GH9aqpdaaeWaqaaiaahMfadaqhaaWcbaGaamiAaaqaamaabmaabaGa amiDaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaaabaGaamiAaiabg2 da9iaaigdaaeaacaWGibaaniabggHiLdGccqGH9aqpcaWHzbWaaWba aSqabeaadaqadaqaaiaadshacqGHsislcaaIXaaacaGLOaGaayzkaa aaaOGaaiOlaaaa@5559@

    References

    Australian Bureau of Statistics (ABS) (2012a). Australian demographic statistics, June Quarter 2012, Catalogue Number 3101.0.

    Australian Bureau of Statistics (ABS) (2012b). Business indicators, June Quarter 2012, Catalogue Number 5676.0.

    Australian Bureau of Statistics (ABS) (2012c). Counts of Australian businesses, including entries and exits, June 2008 to June 2012, Catalogue Number 8165.0.

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    Bell, P. (1999). Comparison of alternative LFS estimators – MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqa aaaaaaaaWdbiaa=nbiaaa@37A3@ Issues for discussion. Methodology Advisory Committee, October 1999.

    Bell, P. (2001). Comparison of alternative Labour Force Survey estimators. Survey Methodology, 27, 1, 53-63.

    Fuller, W.A., and Rao, J.N.K. (2001). A regression composite estimator with application to the Canadian Labour Force Survey. Survey Methodology, 27, 1, 45-51.

    Gambino, J., Kennedy, B. and Singh, M.P. (2001). Regression composite estimation for the Canadian Labour Force Survey: Evaluation and implementation. Survey Methodology, 27, 1, 65-74.

    Gurney, M., and Daly, J.F. (1965). A multivariate approach to estimation in periodic sample surveys. Proceedings of the Section of Survey Research Methods, American Statistical Association, 242-257.

    Särndal, C.-E., Swenson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. New York: Springer-Verlag.

    Singh, A.C. (1994). Sampling design based estimating functions for finite population totals. Invited paper, Abstracts of the Annual Meeting of the Statistical Society of Canada, Banff, Alberta, May 8-11, 48.

    Singh, A.C. (1996). Combining information in survey sampling by modified regression. Proceedings of the Section of Survey Research Methods, American Statistical Association, 120-129.

    Singh, A.C., and Merkouris, P. (1995). Composite estimation by modified regression for repeated surveys. Proceedings of the Section of Survey Research Methods, American Statistical Association, 420-425.

    Singh, A.C., Kennedy, B. and Wu, S. (2001). Regression composite estimation for the Canadian Labour Force Survey with a rotating panel design. Survey Methodology, 27, 1, 33-44.

    Singh, A.C., Kennedy, B., Wu, S. and Brisebois, F. (1997). Composite estimation for the Canadian Labour Force Survey. Proceedings of the Section of Survey Research Methods, American Statistical Association, 300-305.

    Yansaneh, I.S., and Fuller, W.A. (1998). Optimal recursive estimation for repeated surveys. Survey Methodology, 24, 1, 31-40.

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