A short note on quantile and expectile estimation in unequal probability samples 2. Quantile estimation

We consider a finite population with N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@37DF@  elements and a continuous survey variable Y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaiaac6 caaaa@389C@  We are interested in quantiles of the cumulative distribution function F ( y ) = i = 1 N 1 { Y i y } / N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamyEaaGaayjkaiaawMcaaiaai2dadaaeWaqabSqaaiaadMga caaI9aGaaGymaaqaaiaad6eaa0GaeyyeIuoakmaalyaabaGaaGymam aacmaabaGaamywamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadMha aiaawUhacaGL9baaaeaacaWGobaaaaaa@48FF@  and define as

Q ( α ) inf { arg min q i = 1 N w α ( Y i q ) | Y i q | } ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyuamaabm aabaGaeqySdegacaGLOaGaayzkaaGaaGypaiaabMgacaqGUbGaaeOz amaacmaabaGaciyyaiaackhacaGGNbWaaCbeaeaaciGGTbGaaiyAai aac6gaaSqaaiaadghaaeqaaOWaaabCaeqaleaacaWGPbGaaGypaiaa igdaaeaacaWGobaaniabggHiLdGccaaMe8Uaam4DamaaBaaaleaacq aHXoqyaeqaaOWaaeWaaeaacaWGzbWaaSbaaSqaaiaadMgaaeqaaOGa eyOeI0IaamyCaaGaayjkaiaawMcaamaaemaabaGaaGPaVlaadMfada WgaaWcbaGaamyAaaqabaGccqGHsislcaWGXbGaaGPaVdGaay5bSlaa wIa7aaGaay5Eaiaaw2haaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8 UaaiikaiaaikdacaGGUaGaaGymaiaacMcaaaa@6C5B@

the Quantile function of Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37EA@ (see Koenker 2005), where

w α ( ε ) = ( α for ε > 0 1 α for ε 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacqaHXoqyaeqaaOWaaeWaaeaacqaH1oqzaiaawIcacaGLPaaa caaI9aWaaeqaaeaafaqaaeGacaaabaGaeqySdegabaGaaeOzaiaab+ gacaqGYbGaaGjbVlabew7aLjaai6dacaaIWaaabaGaaGymaiabgkHi Tiabeg7aHbqaaiaabAgacaqGVbGaaeOCaiaaysW7cqaH1oqzcqGHKj YOcaaIWaGaaGOlaaaaaiaawUhaaaaa@5492@

The “inf” argument in (2.1) is required in finite populations since the “arg min” is not unique. We draw a sample from the population with known inclusion probabilities π i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@3A9D@ i = 1, , N . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyAaiaai2 dacaaIXaGaaGilaiablAciljaaiYcacaWGobGaaiOlaaaa@3D8F@ Denoting by y 1 , , y n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaaIXaaabeaakiaaiYcacqWIMaYscaaISaGaamyEamaaBaaa leaacaWGUbaabeaaaaa@3DA6@ the resulting sample, we estimate the quantile function by replacing (2.1) through its weighted sample version

Q ^ N ( α ) = inf { arg min q j = 1 n 1 π j w α , j | y j q | } ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyuayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqaHXoqyaiaawIcacaGL PaaacaaI9aGaciyAaiaac6gacaGGMbWaaiWaaeaaciGGHbGaaiOCai aacEgadaWfqaqaaiGac2gacaGGPbGaaiOBaaWcbaGaamyCaaqabaGc daaeWbqabSqaaiaadQgacaaI9aGaaGymaaqaaiaad6gaa0GaeyyeIu oakiaaysW7daWcaaqaaiaaigdaaeaacqaHapaCdaWgaaWcbaGaamOA aaqabaaaaOGaam4DamaaBaaaleaacqaHXoqycaaISaGaamOAaaqaba GcdaabdaqaaiaaykW7caWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyOe I0IaamyCaiaaykW7aiaawEa7caGLiWoaaiaawUhacaGL9baacaaMf8 UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaikda caGGPaaaaa@6DA0@

with w α , j = w α ( y j q ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacqaHXoqycaaISaGaamOAaaqabaGccaaI9aGaam4DamaaBaaa leaacqaHXoqyaeqaaOWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaae qaaOGaeyOeI0IaamyCaaGaayjkaiaawMcaaaaa@44A9@ as defined above. It is easy to see that the sum in (2.2) is a design-unbiased estimate for the sum in Q ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyuamaabm aabaGaeqySdegacaGLOaGaayzkaaaaaa@3B0A@ given in (2.1). Nonetheless, because we take the “arg min” it follows that Q ^ N ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyuayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqaHXoqyaiaawIcacaGL Paaaaaa@3C23@ is not unbiased for Q ( α ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyuamaabm aabaGaeqySdegacaGLOaGaayzkaaGaaiOlaaaa@3BBC@ We therefore look at consistency statements for Q ^ N ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyuayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqaHXoqyaiaawIcacaGL Paaaaaa@3C23@ as follows. Let R i ( q ) = w α ( y i q ) | y i q | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyCaaGaayjkaiaawMcaaiaa i2dacaWG3bWaaSbaaSqaaiabeg7aHbqabaGcdaqadaqaaiaadMhada WgaaWcbaGaamyAaaqabaGccqGHsislcaWGXbaacaGLOaGaayzkaaWa aqWaaeaacaaMc8UaamyEamaaBaaaleaacaWGPbaabeaakiabgkHiTi aadghacaaMc8oacaGLhWUaayjcSdaaaa@4EE9@ and

R ¯ N ( q ) := 1 N i R i ( q ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOuayaara WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWGXbaacaGLOaGaayzk aaGaaGOoaiaai2dadaWcaaqaaiaaigdaaeaacaWGobaaamaaqafabe WcbaGaamyAaaqab0GaeyyeIuoakiaaysW7caWGsbWaaSbaaSqaaiaa dMgaaeqaaOWaaeWaaeaacaWGXbaacaGLOaGaayzkaaGaaGOlaaaa@4886@

We draw a sample from R i ( q ) , i = 1, , N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyCaaGaayjkaiaawMcaaiaa iYcacaWGPbGaaGypaiaaigdacaaISaGaeSOjGSKaaGilaiaad6eaaa a@420D@ and assume we apply a consistent sampling scheme in that

r ¯ n ( q ) := 1 N j = 1 n 1 π j r j ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGXbaacaGLOaGaayzk aaGaaGOoaiaai2dadaWcaaqaaiaaigdaaeaacaWGobaaamaaqahabe WcbaGaamOAaiaai2dacaaIXaaabaGaamOBaaqdcqGHris5aOGaaGPa VpaalaaabaGaaGymaaqaaiabec8aWnaaBaaaleaacaWGQbaabeaaaa GccaWGYbWaaSbaaSqaaiaadQgaaeqaaOWaaeWaaeaacaWGXbaacaGL OaGaayzkaaaaaa@4E6F@

is design-consistent for R ¯ N ( q ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOuayaara WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWGXbaacaGLOaGaayzk aaGaaiilaaaa@3C33@ where r j ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGQbaabeaakmaabmaabaGaamyCaaGaayjkaiaawMcaaaaa @3BA7@ denotes the sample of R i ( q ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyCaaGaayjkaiaawMcaaiaa c6caaaa@3C38@ Note that r j ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGQbaabeaakmaabmaabaGaamyCaaGaayjkaiaawMcaaaaa @3BA7@ and hence r ¯ n ( q ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGXbaacaGLOaGaayzk aaGaaiilaaaa@3C73@ R i ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyCaaGaayjkaiaawMcaaaaa @3B86@ and R ¯ N ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOuayaara WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWGXbaacaGLOaGaayzk aaaaaa@3B83@ also depend on α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@38AB@ which is suppressed in the notation for readability. Let q 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIWaaabeaaaaa@38E8@ be the minimum of R ¯ N ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOuayaara WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWGXbaacaGLOaGaayzk aaaaaa@3B83@ which is not necessarily unique due to the finite structure of the population. We can take the “inf” argument, i.e., q 0 = inf { arg min R ¯ N ( q ) } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIWaaabeaakiaai2daciGGPbGaaiOBaiaacAgadaGadaqa aiaabggacaqGYbGaae4zaiaaykW7ciGGTbGaaiyAaiaac6gaceWGsb GbaebadaWgaaWcbaGaamOtaaqabaGcdaqadaqaaiaadghaaiaawIca caGLPaaaaiaawUhacaGL9baacaGGSaaaaa@4AFC@ but for simplicity we assume a superpopulation model (see Isaki and Fuller 1982) by considering the finite population to be a sample from an infinite superpopulation. In the latter we assume that survey variable Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37EA@ has a continuous cumulative distribution function so q 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIWaaabeaaaaa@38E8@ results in a unique α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@38AB@ quantile. We get for δ > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaaG Opaiaaicdaaaa@3A33@

P ( r ¯ n ( q 0 ) < r ¯ n ( q 0 δ ) ) P ( 1 N j = 1 n 1 π j { r j ( q 0 ) r j ( q 0 δ ) } < 0 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuamaabm aabaGabmOCayaaraWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWG XbWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaGaaGipaiqadk hagaqeamaaBaaaleaacaWGUbaabeaakmaabmaabaGaamyCamaaBaaa leaacaaIWaaabeaakiabgkHiTiabes7aKbGaayjkaiaawMcaaaGaay jkaiaawMcaaiabgsDiBlaadcfadaqadaqaamaalaaabaGaaGymaaqa aiaad6eaaaWaaabCaeqaleaacaWGQbGaaGypaiaaigdaaeaacaWGUb aaniabggHiLdGccaaMc8+aaSaaaeaacaaIXaaabaGaeqiWda3aaSba aSqaaiaadQgaaeqaaaaakmaacmaabaGaamOCamaaBaaaleaacaWGQb aabeaakmaabmaabaGaamyCamaaBaaaleaacaaIWaaabeaaaOGaayjk aiaawMcaaiabgkHiTiaadkhadaWgaaWcbaGaamOAaaqabaGcdaqada qaaiaadghadaWgaaWcbaGaaGimaaqabaGccqGHsislcqaH0oazaiaa wIcacaGLPaaaaiaawUhacaGL9baacaaI8aGaaGimaaGaayjkaiaawM caaiaai6caaaa@6C4E@

Note that the argument in the probability statement is a design-consistent estimate for R ¯ N ( q 0 ) R ¯ N ( q 0 δ ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOuayaara WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWGXbWaaSbaaSqaaiaa icdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0IabmOuayaaraWaaSbaaS qaaiaad6eaaeqaaOWaaeWaaeaacaWGXbWaaSbaaSqaaiaaicdaaeqa aOGaeyOeI0IaeqiTdqgacaGLOaGaayzkaaGaaiilaaaa@4609@ which is less than zero since q 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIWaaabeaaaaa@38E8@ is the minimum of R ¯ N ( ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOuayaara WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqGHflY1aiaawIcacaGL PaaacaGGUaaaaa@3D89@ Hence, the probability tends to one in the sense of design consistency defined in Isaki and Fuller (1982). The same holds of course for δ < 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaaG ipaiaaicdacaGGUaaaaa@3AE3@ With this statement we may conclude that the estimated minimum q ^ 0 = arg min j = 1 n 1 / π j r j ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyCayaaja WaaSbaaSqaaiaaicdaaeqaaOGaaGypaiaabggacaqGYbGaae4zaiaa ykW7ciGGTbGaaiyAaiaac6gadaaeWaqabSqaaiaadQgacaaI9aGaaG ymaaqaaiaad6gaa0GaeyyeIuoakmaalyaabaGaaGymaaqaaiabec8a WnaaBaaaleaacaWGQbaabeaaaaGccaaMe8UaamOCamaaBaaaleaaca WGQbaabeaakmaabmaabaGaamyCaaGaayjkaiaawMcaaaaa@5035@ is a design-consistent estimate for q 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa aaleaacaaIWaaabeaaaaa@38E8@ so that Q ^ N ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyuayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqaHXoqyaiaawIcacaGL Paaaaaa@3C23@ in (2.2) is in turn design-consistent for Q N ( α ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGobaabeaakmaabmaabaGaeqySdegacaGLOaGaayzkaaGa aiOlaaaa@3CC5@ It is easily shown that Q ^ N ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyuayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqaHXoqyaiaawIcacaGL Paaaaaa@3C23@ is the inverse of the normed weighted cumulative distribution function

F ^ N ( y ) := j = 1 n 1 { y j y } / π j j = 1 n 1 / π j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaGaaGOoaiaai2dadaWcaaqaamaaqahabeWcbaGaamOAaiaai2daca aIXaaabaGaamOBaaqdcqGHris5aOGaaGPaVpaalyaabaGaaGymamaa cmaabaGaamyEamaaBaaaleaacaWGQbaabeaakiabgsMiJkaadMhaai aawUhacaGL9baaaeaacqaHapaCdaWgaaWcbaGaamOAaaqabaaaaaGc baWaaabCaeqaleaacaWGQbGaaGypaiaaigdaaeaacaWGUbaaniabgg HiLdGccaaMc8+aaSGbaeaacaaIXaaabaGaeqiWda3aaSbaaSqaaiaa dQgaaeqaaaaaaaaaaa@59EC@

using the same notation as in Kuk (1988). Note that F ^ N ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaaaaa@3B77@ is the Hajek (1971) estimate of the cumulative distribution function (see also Rao and Wu 2009) and as such not a Horvitz-Thompson estimate. As a consequence Q ^ N ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmyuayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqaHXoqyaiaawIcacaGL Paaaaaa@3C23@ is not design-unbiased. Nonetheless, F ^ N ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaaaaa@3B77@ is a valid distribution function, and hence it can be considered as normalized version of the Lahiri or Horvitz-Thompson estimator of the distribution function (see Lahiri 1951) which is denoted by

F ^ L ( y ) := 1 N j = 1 n 1 / π j 1 { y j y } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYeaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaGaaGOoaiaai2dadaWcaaqaaiaaigdaaeaacaWGobaaamaaqahabe WcbaGaamOAaiaai2dacaaIXaaabaGaamOBaaqdcqGHris5aOWaaSGb aeaacaaIXaaabaGaeqiWda3aaSbaaSqaaiaadQgaaeqaaOGaaGymam aacmaabaGaamyEamaaBaaaleaacaWGQbaabeaakiabgsMiJkaadMha aiaawUhacaGL9baaaaGaaiOlaaaa@5075@

Kuk (1988) proposes to replace F ^ L ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYeaaeqaaOWaaeWaaeaacqGHflY1aiaawIcacaGL Paaaaaa@3CC1@ with alternative estimates of the distribution function: Instead of estimating the distribution function itself he suggests to estimate the complementary proportion S ^ R ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabm4uayaaja WaaSbaaSqaaiaadkfaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaaaaa@3B88@ which then leads to the estimate F ^ R ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaaaaa@3B7B@ defined through

F ^ R ( y ) = 1 S ^ R ( y ) = 1 1 N j = 1 n 1 / π j 1 { y j > y } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaGaaGypaiaaigdacqGHsislceWGtbGbaKaadaWgaaWcbaGaamOuaa qabaGcdaqadaqaaiaadMhaaiaawIcacaGLPaaacaaI9aGaaGymaiab gkHiTmaalaaabaGaaGymaaqaaiaad6eaaaWaaabCaeqaleaacaWGQb GaaGypaiaaigdaaeaacaWGUbaaniabggHiLdGcdaWcgaqaaiaaigda aeaacqaHapaCdaWgaaWcbaGaamOAaaqabaGccaaIXaWaaiWaaeaaca WG5bWaaSbaaSqaaiaadQgaaeqaaOGaaGOpaiaadMhaaiaawUhacaGL 9baaaaGaaGOlaaaa@5763@

Resulting directly from these definitions we can express F ^ R ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaOWaaeWaaeaacqGHflY1aiaawIcacaGL Paaaaaa@3CC7@ in terms of F ^ N ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacqGHflY1aiaawIcacaGL Paaaaaa@3CC3@ through

F ^ R = 1 1 N j = 1 n 1 / π j + F ^ L and F ^ L = j = 1 n 1 / π j N F ^ N . ( 2.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaOGaaGypaiaaigdacqGHsisldaWcaaqa aiaaigdaaeaacaWGobaaamaaqahabeWcbaGaamOAaiaai2dacaaIXa aabaGaamOBaaqdcqGHris5aOWaaSGbaeaacaaIXaaabaGaeqiWda3a aSbaaSqaaiaadQgaaeqaaOGaey4kaSIabmOrayaajaWaaSbaaSqaai aadYeaaeqaaaaakiaaywW7caqGHbGaaeOBaiaabsgacaaMf8UabmOr ayaajaWaaSbaaSqaaiaadYeaaeqaaOGaaGypamaalaaabaWaaabCae qaleaacaWGQbGaaGypaiaaigdaaeaacaWGUbaaniabggHiLdGcdaWc gaqaaiaaigdaaeaacqaHapaCdaWgaaWcbaGaamOAaaqabaaaaaGcba GaamOtaaaaceWGgbGbaKaadaWgaaWcbaGaamOtaaqabaGccaaIUaGa aGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6caca aIZaGaaiykaaaa@69D6@

Kuk (1988) shows that, under sampling with unequal probabilities, estimation of the median derived from F ^ R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaaaa@38EA@ outperforms median estimates derived from F ^ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaad6eaaeqaaaaa@38E6@ and F ^ L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYeaaeqaaaaa@38E4@ in terms of mean squared estimation error. Note that the estimators F ^ N , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaad6eaaeqaaOGaaiilaaaa@39A0@ F ^ L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYeaaeqaaaaa@38E4@ and F ^ R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaaaa@38EA@ coincide in the case of simple random sampling without replacement where π j = π = n / N . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadQgaaeqaaOGaaGypaiabec8aWjaai2dadaWcgaqaaiaa d6gaaeaacaWGobaaaiaac6caaaa@3FC7@

Date modified: