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    Survey Methodology

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    2. Generating synthetic populations from single survey data that accounts for complex sampling designs

    Qi Dong, Michael R. Elliott and Trivellore E. Raghunathan

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    Dong et al. (2014) extended work in the finite population Bayesian bootstrap to develop a non-parametric approach to the generation of posterior predictive distributions. A summary of the algorithm to draw the l -th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYgacaqGTa GaaeiDaiaabIgaaaa@395F@  of l = 1 , ... , L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYgacqGH9a qpcaaIXaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGmbaaaa@3CD5@  synthetic populations for stratified, clustered sample designs with unequal probabilities of selection is as follows:

    1. Use the Bayesian Bootstrap (BB) (Rubin 1981) to adjust for stratification and clustering. Draw a simple random sample with replacement (SRSWR) of size m h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyBa8aadaWgaaWcbaWdbiaadIgaa8aabeaaaaa@3835@  from the c h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ya8aadaWgaaWcbaWdbiaadIgaa8aabeaaaaa@382B@  clusters within each stratum h = 1 , ... , H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacqGH9a qpcaaIXaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGibaaaa@3CCD@  and calculate bootstrap replicate weights for each of the n h i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gadaWgaa WcbaGaamiAaiaadMgaaeqaaaaa@38D6@  observations in each cluster as w * ( l ) = { w h i * ( l ) ,     h = 1 , … ,   H ,     i = 1 , … ,   c h ,     k = 1 , … ,   n h i } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaahaaWcbeqaa8qacaGGQaGaaiikaiaadYgacaGGPaaa aOGaeyypa0ZaaiWaa8aabaWdbiaadEhapaWaa0baaSqaa8qacaWGOb GaamyAaaWdaeaapeGaaiOkaiaacIcacaWGSbGaaiykaaaakiaacYca caGGGcGaaiiOaiaadIgacqGH9aqpcaaIXaGaaiilaiabgAci8kaacY cacaGGGcGaamisaiaacYcacaGGGcGaaiiOaiaadMgacqGH9aqpcaaI XaGaaiilaiabgAci8kaacYcacaGGGcGaam4ya8aadaWgaaWcbaWdbi aadIgaa8aabeaak8qacaGGSaGaaiiOaiaacckacaWGRbGaeyypa0Ja aGymaiaacYcacqGHMacVcaGGSaGaaiiOaiaad6gapaWaaSbaaSqaa8 qacaWGObGaamyAaaWdaeqaaaGcpeGaay5Eaiaaw2haaiaacYcaaaa@6814@  where w h i k * = w h i k ( ( 1 − ( m h / c h − 1 ) ) + ( m h / c h − 1 ) ( c h / m h )   m h i * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaqhaaWcbaWdbiaadIgacaWGPbGaam4AaaWdaeaapeGa aiOkaaaakiabg2da9iaadEhapaWaaSbaaSqaa8qacaWGObGaamyAai aadUgaa8aabeaak8qadaqadaWdaeaapeWaaeWaa8aabaWdbiaaigda cqGHsisldaGcaaWdaeaapeWaaeWaaeaadaWcgaqaaiaad2gapaWaaS baaSqaa8qacaWGObaapaqabaaak8qabaGaam4ya8aadaWgaaWcbaWd biaadIgaa8aabeaak8qacqGHsislcaaIXaaaaaGaayjkaiaawMcaaa WcbeaaaOGaayjkaiaawMcaaiabgUcaRmaakaaapaqaa8qadaqadaqa amaalyaabaGaamyBa8aadaWgaaWcbaWdbiaadIgaa8aabeaaaOWdbe aacaWGJbWdamaaBaaaleaapeGaamiAaaWdaeqaaOWdbiabgkHiTiaa igdaaaaacaGLOaGaayzkaaaaleqaaOWaaeWaaeaadaWcgaqaaiaado gapaWaaSbaaSqaa8qacaWGObaapaqabaaak8qabaGaamyBa8aadaWg aaWcbaWdbiaadIgaa8aabeaaaaaak8qacaGLOaGaayzkaaGaaiiOai aad2gapaWaa0baaSqaa8qacaWGObGaamyAaaWdaeaapeGaaiOkaaaa aOGaayjkaiaawMcaaaaa@619D@  and m h i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyBa8aadaqhaaWcbaWdbiaadIgacaWGPbaapaqaa8qacaGGQaaa aaaa@39E2@  denotes the number of times that cluster i ,     i = 1 ,   … ,   c h   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaacYcacaGGGcGaaeiiaiaadMgacqGH9aqpcaaIXaGaaiil aiaacckacqGHMacVcaGGSaGaaiiOaiaadogapaWaaSbaaSqaa8qaca WGObaapaqabaGcpeGaaiiOaaaa@44B3@  is selected. To ensure all the replicate weights are non-negative, m h ≤ ( c h − 1 ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyBa8aadaWgaaWcbaWdbiaadIgaa8aabeaak8qacqGHKjYOdaqa daWdaeaapeGaam4ya8aadaWgaaWcbaWdbiaadIgaa8aabeaak8qacq GHsislcaaIXaaacaGLOaGaayzkaaGaai4oaaaa@405C@  here and below we take m h = ( c h − 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyBa8aadaWgaaWcbaWdbiaadIgaa8aabeaak8qacqGH9aqpdaqa daWdaeaapeGaam4ya8aadaWgaaWcbaWdbiaadIgaa8aabeaak8qacq GHsislcaaIXaaacaGLOaGaayzkaaGaaiOlaaaa@3FA0@
    2. Use the finite population Bayesian bootstrap (FPBB) (Lo 1986; Cohen 1997) for unequal probabilities of selection to adjust for unequal probabilities of selection. For each cluster i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMgaaaa@36CA@  in stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgaaaa@36C9@  of population size N h i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6eadaWgaa WcbaGaamiAaiaadMgaaeqaaOGaaiilaaaa@3970@  draw a sample of size N h i − n h i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadIgacaWGPbaapaqabaGcpeGaeyOe I0IaamOBa8aadaWgaaWcbaWdbiaadIgacaWGPbaapaqabaGccaGGSa aaaa@3DED@  denoted by ( y 1 * , … , y N h i − n h i * ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaadMhapaWaa0baaSqaa8qacaaIXaaapaqaa8qa caGGQaaaaOGaaiilaiabgAci8kaacYcacaWG5bWdamaaDaaaleaape GaamOta8aadaWgaaadbaWdbiaadIgacaWGPbaapaqabaWcpeGaeyOe I0IaamOBa8aadaWgaaadbaWdbiaadIgacaWGPbaapaqabaaaleaape GaaiOkaaaaaOGaayjkaiaawMcaaiaacYcaaaa@4775@  by drawing y h i k * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaqhaaWcbaWdbiaadIgacaWGPbGaam4AaaWdaeaapeGa aiOkaaaaaaa@3ADE@  from cluster data ( y 1 , … , y n h i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaadMhapaWaaSbaaSqaa8qacaaIXaaapaqabaGc peGaaiilaiabgAci8kaacYcacaWG5bWdamaaBaaaleaapeGaamOBa8 aadaWgaaadbaWdbiaadIgacaWGPbaapaqabaaaleqaaaGcpeGaayjk aiaawMcaaaaa@4156@  with probability w h i k * − 1 + l h i k , j − 1 * ( N h i − n h i ) / n h i N c H − n c H + ( j − 1 ) * ( N h i − n h i ) / n h i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadEhapaWaa0baaSqaa8qacaWGObGaamyAaiaa dUgaa8aabaWdbiaacQcaaaGccqGHsislcaaIXaGaey4kaSIaamiBa8 aadaWgaaWcbaWdbiaadIgacaWGPbGaam4AaiaacYcacaWGQbGaeyOe I0IaaGymaaWdaeqaaOWdbiaacQcadaWcgaqaamaabmaapaqaa8qaca WGobWdamaaBaaaleaapeGaamiAaiaadMgaa8aabeaak8qacqGHsisl caWGUbWdamaaBaaaleaapeGaamiAaiaadMgaa8aabeaaaOWdbiaawI cacaGLPaaaaeaacaWGUbWdamaaBaaaleaapeGaamiAaiaadMgaa8aa beaaaaaakeaapeGaamOta8aadaWgaaWcbaWdbiaadogapaWaaSbaaW qaa8qacaWGibaapaqabaaaleqaaOWdbiabgkHiTiaad6gapaWaaSba aSqaa8qacaWGJbWdamaaBaaameaapeGaamisaaWdaeqaaaWcbeaak8 qacqGHRaWkdaqadaWdaeaapeGaamOAaiabgkHiTiaaigdaaiaawIca caGLPaaacaGGQaWaaSGbaeaadaqadaWdaeaapeGaamOta8aadaWgaa WcbaWdbiaadIgacaWGPbaapaqabaGcpeGaeyOeI0IaamOBa8aadaWg aaWcbaWdbiaadIgacaWGPbaapaqabaaak8qacaGLOaGaayzkaaaaba GaamOBa8aadaWgaaWcbaWdbiaadIgacaWGPbaapaqabaaaaaaak8qa caGGSaaaaa@6C98@  where w h i k * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaqhaaWcbaWdbiaadIgacaWGPbGaam4AaaWdaeaapeGa aiOkaaaaaaa@3ADC@  is the replicate weight of unit k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgaaaa@36CC@  in cluster i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaaaa@36EA@  in stratum h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaGGSa aaaa@3779@  and l h i k , j − 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiBa8aadaWgaaWcbaWdbiaadIgacaWGPbGaam4AaiaacYcacaWG QbGaeyOeI0IaaGymaaWdaeqaaaaa@3D59@  is the number of bootstrap selections of y h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaadIgacaWGPbGaam4AaaWdaeqaaaaa @3A1F@  among y 1 * , … , y j − 1 * . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaqhaaWcbaWdbiaaigdaa8aabaWdbiaacQcaaaGccaGG SaGaeyOjGWRaaiilaiaadMhapaWaa0baaSqaa8qacaWGQbGaeyOeI0 IaaGymaaWdaeaapeGaaiOkaaaak8aacaGGUaaaaa@413F@  Form the FPBB population y 1 , … , y n h i ,   y 1 * , … , y N h i − n h i * . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaeyOj GWRaaiilaiaadMhapaWaaSbaaSqaa8qacaWGUbWdamaaBaaameaape GaamiAaiaadMgaa8aabeaaaSqabaGcpeGaaiilaiaacckacaWG5bWd amaaDaaaleaapeGaaGymaaWdaeaapeGaaiOkaaaakiaacYcacqGHMa cVcaGGSaGaamyEa8aadaqhaaWcbaWdbiaad6eapaWaaSbaaWqaa8qa caWGObGaamyAaaWdaeqaaSWdbiabgkHiTiaad6gapaWaaSbaaWqaa8 qacaWGObGaamyAaaWdaeqaaaWcbaWdbiaacQcaaaGcpaGaaiOlaaaa @5164@
    3. Produce F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOraaaa@36C7@  FPBB samples for each BB sample, denoted by S l 1 ,   … ,   S l F ,   l = 1 , … , L . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaadYgacaaIXaaapaqabaGcpeGaaiil aiaacckacqGHMacVcaGGSaGaaiiOaiaadofapaWaaSbaaSqaa8qaca WGSbGaamOraaWdaeqaaOWdbiaacYcacaGGGcGaamiBaiabg2da9iaa igdacaGGSaGaeyOjGWRaaiilaiaadYeacaGGUaaaaa@4A29@  Pool the F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOraaaa@36C7@  FPBB samples to produce one synthetic population, S l . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4ua8aadaWgaaWcbaWdbiaadYgaa8aabeaakiaac6caaaa@38D9@  (Because N = ∑ h ∑ i N h i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6eacqGH9a qpdaaeqaqaamaaqababaGaamOtamaaBaaaleaacaWGObGaamyAaaqa baaabaGaamyAaaqab0GaeyyeIuoaaSqaaiaadIgaaeqaniabggHiLd aaaa@4025@  may be unrealistically large, generating a sample of size k * n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4AaiaacQcacaWGUbaaaa@388D@  for large k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgaaaa@36CC@  is sufficient.)

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