Impact of Measurement Errors on Estimators of Parameters of a Finite Population with Linear Trend Under Systematic Sampling

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1 Amec Joul of Theoetcl d Appled Stttc 07; 6(6): do: 0.648/.t ISSN: (Pt); ISSN: (Ole) Impct of Meuemet Eo o Etmto of Pmete of Fte ulto w Le Ted de Stemtc Smplg Oloo Odhmbo Eck *, Jme Kh, Wful Mke Eck Deptmet of Stttc d Actul Scece, Kett vet (K), Nob, Ke Eml dde: Ecoloo58@gml.com (O. O. Eck) * Coepodg uo To cte tcle: Oloo Odhmbo Eck, Jme Kh, Wful Mke Eck. Impct of Meuemet Eo o Etmto of Pmete of Fte ulto w Le Ted de Stemtc Smplg. Amec Joul of Theoetcl d Appled Stttc. Vol. 6, No. 6, 07, pp do: 0.648/.t Receved: Septembe 7, 07; Accepted: Octobe 4, 07; Publhed: Novembe 0, 07 Abtct: The tud volve vetgtg e mpct of meuemet eo o etmto of pmete of fte populto w le ted mog populto vlue, ude temtc mplg. The tud povde deep udetdg o e mout d tue of devto toduced b eo d how ee eo ffect etmto of pmete of populto w le ted. Codeto gve to meuemet eo t ume oml dtbuto. Stemtc mplg techque ued whee mple of ze elected doml fom fte populto w fxed tevl. Stemtc mplg codeed ted of mple dom mplg ce becue of t effectvee delg w le ted. The explct vlue of populto totl, me d vce togee w e etmte e deved. The eult dcte t ee c be oveetmte of e populto me f e expected temtc eo ted towd potve vlue d udeetmte f e expected temtc eo ted towd egtve vlue. Whe dom eo e codeed, ee o effect o etmted populto pmete. Kewod: Fte ulto w Le Ted, Stemtc Smplg, Meuemet Eo. Itoducto I del tuto, t umed t ough ome kd of pobblt mplg, ce temtc mplg, e obevto o e ut e coect vlue fo t ut, d t mplg eo m e olel fom e dom mplg vto t peet whe ut e meued ted of complete populto of N ut. Cot to e umpto, o-mplg eo t e due to meuemet o obevto do occu t dt collecto tge. The tue coe eo good mple model fo meuemet. It cot of tue vlue d two eo compoet; dom eo d temtc eo. Whee, = t + e + ε () e meued vlue t e tue vlue e e dom eo ε e temtc eo Rdom eo cued b upedctble fluctuto e edg of meuemet pptu o expemete tepetto of e tumetl edg. Stemtc eo cued b phcl fcto t ffect expemet o meuemet of e vble co e mple pedctble decto. Accodg to Fulle d Coll et l [, ], t well kow t meuemet eo obeved dt c led eeche to dw coect feece. Much of e el wok e focued o e tpcl textbook model of clcl meuemet eo. Boud et l [3] cocluded e uve of meuemet eo b cllg fo eeche to p moe tteto to e poblt of o-clcl meuemet eo, bo cceg e lkel be e le t tke o

2 7 Oloo Odhmbo Eck et l.: Impct of Meuemet Eo o Etmto of Pmete of Fte ulto w Le Ted de Stemtc Smplg ccout of meuemet eo d devg pocedue t coect uch eo. I ecet e, umbe of ppe hve exmed e coequece of o-clcl meuemet eo lbo ecoomc. [4-6], ll oted t o-clcl meuemet eo of e tpe tpcll foud come dt, tteute e ole of whte oe meuemet eo model of eg dmc. Meuemet eo c bet be tuded f e tue vlue obted. Th ppoch lmted to tem fo whch feble meod fo fdg e tue vlue ext. Fo tce mot of tude ug Bod M Idex(BMI) el o elf-epoted meue fom uve dt et. Howeve Coo et l how t ee lge bod of evdece whch ugget t elf-epoted BMI ted to udeetmte tue BMI; occu bo becue people udeepot e weght d oveetmte e heght [7]. Lookg t meuemet eo elf-epoted BMI pecfcll, Plke et l exme e coequece of ee eo whe clfg people ccodg to obet ttu [8]. Stommel et l [9] comped elf-epoted d ecoded BMI ug S dt d foud ubttl mout of mclfcto of obet ttu whe ug elf-epoted BMI, ptcull e exteme (oveweght o udeweght) ctegoe. Coequece of meuemet eo wee exmed whe lg e mpct of BMI o ge of hel k. Belloc [0], comped dt o hoptlzto epoted houehold tevew w e hoptl ecod fo e dvdul. Hoptl ecod poduce e tue vlue whch e e comped to e obeved vlue fom houehold tevew. G [] comped emploee' ttemet of ck leve w e peol offce ecod. The compo of dt w to deteme e peece of meuemet eo f. [-3] compe epodet lle w ee docto ecod o e epodet o w e eult of complete medcl exmto. Accodg to Sädl meuemet eo e dug dt collecto tge, d m hve codeble mpct o e etmte [4]. I ecet tude; Nbwg [5] tuded Effect of meuemet eo o populto dom ode. Roell et l [6] tuded e fluece of meuemet eo o clbto, dcmto d ovell etmto of k pedcto model. O'Nel et l [7] exmed e coequece of meuemet eo elf-epoted BMI whe etmtg e eltohp betwee obet d come. Subm et l [8] tuded Etmto of ulto Me e Peece of Le Ted mog populto vlue ude ccul temtc mplg. Ouko et l [9] tuded e effect of Meuemet Eo o ulto Etmte fom Smple Geeted fom Sttfed ulto ough Stemtc Smplg Techque. Gellet et l [0] exmed e Effect of dom eo o dgotc ccuc llutted w opometc dgo of mlutto. I poect fue tud bed o The Impct of Meuemet Eo o Etmto of Pmete fo fte ulto w Le Ted ude Stemtc Smplg. Fte populto w le ted cot of N ut detfed b e lbel,,..., N odeed ceg ze. Though temtc mplg, e populto e dvded to mple of ut ech. The tble below how et of ll poble mple. Tble. Set of ll poble mple. Smple,... h h... vlue + + +h (-)+ (-)+ (-)+h Smple totl t t... t h... t Let temtc mplg be deoted SY. The et of ll poble mple deoted S cot of dffeet et t e o-ovelppg t c be obted S =(...,..., ) () Smplg deg of SY u gve ( ) p = f belogtos 0 oewe Ech mple o clute elected w pobblt d obeved completel pe e deg.. Pmete Etmto.. Etmto of Pmete fo Fte ulto w Le Ted de Stemtc Smplg Let N be e ze of fte populto. Suppoe e fte populto uch t e obeved vlue ume hpoetcl ted = µ + β (3) whee µ d β e cott d =,,... N e odeed ceg ze of e lbel The populto e d to poe le ted mog t vlue. Let populto ze N be multple of, N =. The etmte of populto me ude le temtc mplg whch ce gle dom tt tke obted. N Y = N= = + N = N + = µ + β N ( µ β) (4) Accodg to Mukhopdh [0], populto w le ted h temtc mple gve b µ + β + ( l ), whee e dom tt,

3 Amec Joul of Theoetcl d Appled Stttc 07; 6(6): l =,..., The mple totl (5) = µ β { ( ) } t = + + l The e mple me wtte = µ + β + e me of temtc mple d ough pobblt mplg, t e ubed etmto of Y Smll, populto totl deote tθ = Sce e teet etmtg populto totl, fom deg-bed ppoch, Hovtz-Thompo etmto, HTE, Hovtz d Thompo [] ued. The etmto defed, I tπ = π = π (6) (7) Whee π 0 e ft ode cluo pobblt. de Stemtc Smplg, SY deg w mplg tevl, d e epoe vble, e populto totl etmto fo π = p( I = ) = gve Whee p( I ) t π = = t (8) π π = = = e pobblt t e ut of e populto cluded e mple. Fte populto vce gve ( Y ) N S = N = N ( N + ) = β Vce of me fom mple dom mple gve V = ( ) ( )( + ) N = N de temtc mplg, e vce of me gve V ( ) β ( ) = (9) Accodg to Dog et l [3], fo emovg effect of le β ted, temtc mplg much moe effcet mple dom mplg... Etmto of Vce fom Sgle Stemtc Smple Accodg to Sädl [4], mo dwbck of SY t ee o ubed etmto fo e vce of e etmto of populto me except fo ome ce of ccul temtc mplg. Th becue SY equvlet to clute mplg w ol oe clute elected. Howeve, ude ome umpto bout e tue of e populto, t poble to popoe etmto t e ppoxmtel ubed fo e deg vce. I ce, e ppopte vce etmto fo e populto w le ted whch ce vlue of e ut e tedl ceg b cott mout codeed. M bed etmto hve bee popoed fo kd of populto; Wolte [], mde ltcl tude o populto w le ted d popoed e followg etmto fo e vce of e etmto of populto totl. Aume = m. Sce temtc mple c be looked upo goupg e populto m goup d choog ut fom ech goup of ze, etmto of e me of e g goup w e vce etmto g g =, g =,... m g g v = Hece etmto of V ( t π ) m v = N V g g m f g = ( g g ) v = N, f = N (0) Coch [5], uggeted e etmto below to be ppopte fo ( π ) v = V t = N Whee S ' = = l = temtc mple. ( ) 6( ) ( ) 6( ) e vce of

4 73 Oloo Odhmbo Eck et l.: Impct of Meuemet Eo o Etmto of Pmete of Fte ulto w Le Ted de Stemtc Smplg e mplg tevl. The etmte bed o ucceve qudtc tem e equece. The fcto 6 e um of que of e ' coeffcet e dffeece. The tem e um of que of e weght w. ' due to e weght d e lt mple vlue. ' le mll,. Thu ( π ) ( ) ppled to e ft c be eplced b e fcto ( ) 6( ) = v = V t = N, () fo Yte [6], uggeted e followg etmto mog oe bed uccevel o ecod d hghe ode dffeece. f v3 = N ( 4) = + + () The um of que cot (-4) tem..3. ulto Totl Etmto d It Vce Peece of Rdom Eo Though meuemet pocedue, e dvdul obeved ccomped b e dom eo tem e. The obeved vlue u gve The model c u be expeed = µ + β + e (3) = µ + e (4) Whee - e obeved vlue fo e dvdul, µ + β = µ - e tue vlue fo e dvdul, e - e dom eo fo e dvdul. Fo e tue fucto vlue µ, populto totl Defe t θ = µ N I tπ = = = π π e populto totl etmto (Hovtz-Thompo etmto), w π 0 I SY deg, π = p ( I = ) = The ot expectto of e populto totl etmto obted ; ( ) ( ) E t = E E t = pm π p m π µ The totl vce of t π w epect to mplg deg p (.) d e meuemet model m ccodg to Sädl [4] gve ( π ) = ( π ) + ( π ) Vpm t E p Vm t V p Em t (5) The vce of populto etmto e um of e expected vlue of e codtol vce d e vce of codtol expected vlue. Theefoe totl vce cot of meuemet vce d mplg vce epectvel. Meuemet vce whe decompoed expeed follow; ( ) = + E V tπ δ δ p m Smplg vce whe decompoed lo expeed ( ) ( ) V E tπ = µ µ p m Combg e eult we hve ( ) = + + ( ) V tπ δ δ µ µ pm (6) Th e deved equto fo totl vce peece of dom eo w epect to mplg deg p (.) d meuemet model m otl. The meuemet vce δ + δ h mple epoe vce d e coelted epoe vce t compoet epectvel. ( ) µ µ e mplg vce. Smple epoe vce eflect e dom vto epodet we to uve ove epeted meuemet. Coelted epoe vce lo kow tevewe vce occu becue epoe eo e coelted fo mple ut tevewed b e me tevewe..4. Memtcl Model fo Eo of Meuemet Suppoe meuemet could be depedetl epeted m tme o ut, we could geete dffeet -vlue. Let be e elzed vlue e epeted obevto, e = µ + β + e + ε

5 Amec Joul of Theoetcl d Appled Stttc 07; 6(6): Thu Let µ + β = µ µ e tue vlue of ut Bo e d = µ + e + ε (7) ε e dom vble whee e ( ) e depedet dom vble w ( 0) E e = d vce δ e. ε,... ε e depedet d detcll dtbuted dom vble w e expected vlue ε d vce The dom vble (,... ) dom vble e ( ) δ ε ε = e depedet of e Fom e elected mple cotg of -vlue, e ft d ecod-ode momet e; ( ) ( ) ( ) ( ) ( ) = δ + δ = δ E = E µ + E e + E ε = µ + ε = α m V m e ε ( ) ( ε, ε ) Cov = C = δ m m ε e temtc eo o e dvdul meued vlue E ε = ε e b meuemet. d ( ) lke dom eo, temtc eo ted to be cotetl ee potve o egtve - becue of, temtc eo ometme codeed to be b meuemet..5. Meuemet B d Expectto of π-etmto Meuemet b e whe expected meuemet vlue o elemet do ot gee w tue elemet vlue. pm ( ) pm ( ) B t = E t t π π θ.6. Decompog Vce of ulto Totl Etmto Whe Stemtc Eo Ae Peet Accodg to Sädl [4] e totl vce of t π w epect to mplg deg p (.) d e meuemet model m gve ( π ) = ( π ) + ( π ) Vpm t E p Vm t V p Em t (9) Meuemet vce whe decompoed expeed follow; ( ) = + E V tπ δ δ p m Smll, mplg vce decompoed follow Now ( π ) = ( ) ( ) = ( ) = Vp Em t Vp Em E E α m m ( ) V p Em = V p α ( α ) ( ) (0) ( ) α α cov( ( ) ( )) ( ) ( ) = V I + I I π π α π ππ αα ( ) ( ) α α α ( ) α ( ) ( ) α α, p = + = + = + = α α ( π ) ( ), V E t = α α p m The deved expected meuemet vlue expeed E pm ( ) t = t + π θ ε Combg e eult e totl vce become ( ) = + + ( ) V tπ δ δ α α pm, () The deve totl meuemet b w epect to mplg p. d meuemet model m epectvel u deg ( ) expeed follow B t = t + ε t = ε (8) pm ( ) π θ θ We let α = µ + εk d α = µ + εl Subttutg 7 &8 to 6 bove ( π ) = δ + ( )( )( ) δ + µ + ε µ + ε, δ δ ( ) ( µ µ µ ε l µ εk εkεl ) V t pm k l = , ( π ) = δ + δ + ( ) ( µ µ ) + ( ) ( µ ε + µ ε + ε ε ) V t pm,, l k k l cot of The tem ( ) ( µ ε l + µ εk + εkεl ), temtc eo d t cue vto totl vce. ()

6 75 Oloo Odhmbo Eck et l.: Impct of Meuemet Eo o Etmto of Pmete of Fte ulto w Le Ted de Stemtc Smplg 3. Numecl Reult d Dcuo A fte populto of ze N geeted fo populto wout eo, populto w dom eo d populto w temtc eo. The populto totl vce, e populto me d e populto totl e e computed. I e electo of temtc mple of ze, dom tt elected betwee d cluve whch ce e mplg tevl. To etmte pmete, multo of dt doe 0 tme ech ce e etmte obted. The eult e e veged to get e etmte of ll pmete equed e tud. Etmto of vce doe ug e ee etmto Ce Let N=800, =3, =5 below mplfed to eflect temtc mplg. v v 3 v = (3) ( ) ( ) g= g g ( ) + + = + ( ) fo ( ) 6( ) = N (4) = ( ) fo ( ) (5) 3.5( 4) = N The tble fom ce to ce 3 cot of pmete d e etmte fo populto wout eo d populto w eo. Tble. ulto Pmete d The Etmte fo e N ( 0,) d ε N ( ) 0.6,.5. Pmete Etmte Me Totl.v Etmted me Etmted totl V V V 3 No eo , 80 36, 69, , 980 5, 4, W e , 84 36, 76, , 880 5, 556, 994 8, 875 8, 956 W ε , 35 37, 594, , 03 6, 07, , 734 5, 305 Tble 3. ulto Pmete d The Etmte fo e N ( 0,) d ε N ( ) 0.5,.5. Pmete Etmte Me Totl.v Etmted me Etmted totl V V V 3 No eo , 80 36, 69, , 980 5, 4, W e , 84 36, 883, , 60 5, 66, 97 7, 65 8, 70 W ε , 6 36, 740, , 6 5, 450, 58 44, , 078 Fom tble & 3, e eult how t:. The populto me udeetmted f populto w temtc eo h egtve temtc b.. The populto me oveetmted f populto w temtc eo h potve temtc b. 3. ulto me peece of dom eo lmot Ce Let N=800, =40, =0 cofom to e me of populto wout eo. 4. Etmto of vce of populto totl etmto udeetmte e populto totl vce. 5. v d v 3 fo populto w tue vlue e zeo but ve mll vlue fo populto w eo. Tble 4. ulto Pmete d The Etmte fo e N ( 0,) d ε N ( ) 0.6,.5. Pmete Etmte Me Totl.v Etmted me Etmted totl V V V 3 No eo , 80 3, 46, , 500, 744, W e , 84 3, 645, , 976, 755, 957 6, 807 7, 7 W ε , 35 3, 446, , 68, 87, , 835 6, 698 Tble 5. ulto Pmete d The Etmte fo e N ( 0,) d ε N ( ) 0.5,.5. Pmete Etmte Me Totl.v Etmted me Etmted totl V V V 3 No eo , 80 3, 46, , 550, 744, W e , 84 5, 54, , 70, 706, 4 7, 79 8, 9 W ε , 6 3, 49, , 53, 8, , , 006

7 Amec Joul of Theoetcl d Appled Stttc 07; 6(6): Fom tble 4 & 5, e eult how t:. The populto me udeetmted f e expectto of temtc eo egtve.. The populto me oveetmted f e expectto of temtc eo potve. 3. Etmte fom v fo populto w temtc eo Ce 3 Let N=800, =00, =8 exceed e coepodg etmte fom e populto wout eo. 4. Etmte fom v exceed etmte fom v d v The me of populto w dom eo cloe to e ctul populto me. Tble 6. ulto Pmete d The Etmte fo e N ( 0,) d ε N ( ) 0.6,.5. Pmete Etmte Me Totl.v Etmted me Etmted totl V V V 3 No eo , 80 3, 764, , 6 694, W e , 84 3, 663, , , 97 6, 509 6, 787 W ε , 35 3, 680, , 6 73, 733 9, 70 9, 937 Tble 7. ulto Pmete d The Etmte fo e N ( 0,) d ε N ( ) 0.5,.5. Pmete etmte Me Totl.v Etmted me Etmted totl V V V 3 No eo , 80 3, 764, , , W e , 84 3, 66, , 5 69, 40 6, 887 7, 490 W ε , 6 3, 677, , , 750 8, 466 6, 87 Fom tble 6 & 7, e eult how t:. Whe e mple ze ceed, bo e populto vce d etmted vce e educed.. Etmte fom v e much hghe e epectve etmte fom v d v Potve expected temtc eo oveetmte populto me d totl whle egtve expected temtc eo udeetmte populto me d totl. 4. Summ Fom e tud, t obeved t:. The populto me d hece e populto totl e oveetmted fo e ce whee expectto of temtc eo potve.. The populto me d hece e populto totl e udeetmted fo e ce whee expectto of temtc eo egtve. 3. Impct of dom eo o populto me d populto totl mml d cotet. 4. The vce of populto totl etmto e ll udeetmted ug e ee etmto, v, v d v Icee mple ze led to decee etmted vce of populto totl etmto. 6. Fo populto w temtc eo, e etmted vce e ove epeeted. Etmto v gve hghe vce etmto v d v Cocluo The tud h how t: Impct of dom eo o populto me, populto totl d etmted vce of populto totl etmto ve mml. Stemtc eo poduce temtc b t oveetmte e populto me whe e b potve d udeetmte e populto me whe e b egtve. All e ee etmto udeetmte populto vce d eefoe e e bed. Amog e ee, v bette becue t gve vlue cloe to e populto vce. Geell temtc eo led to ove epeetto of e etmted vce whle dom eo hve o mpct o etmte of populto vce. Refeece [] Fulle, W. (987). Meuemet Eo Model. Wle d So. [] Coll, R., J, R. D. d Stefk, L. (994). Meuemet Eo ole model, Chpm d Hll, Lodo. [3] Boud, J., Bow, C., d Mowetz, N (00). Meuemet Eo Suve dt. Amec Joul of Theoetcl d Appled Stttc, 5. [4] Pchke, J. (995). Meuemet eo d eg dmc: ome etmte fom e pd vldto tud. Joul Bue of Ecomc Stttc, 3(3): [5] O Nel, D., Sweetm, O., d V de ge, D. (007). The effect of meuemet eo d omtted vble whe ug tto mtce to meue tegeetol moblt. Joul of Ecoomc Ieqult, 5(): [6] Gottchlk, P. d Huh, M. (00). Ae eg eqult d moblt ovetted? The Impct of Noclcl Meuemet Eo. Revew of Ecoomc d Stttc, 9():

8 77 Oloo Odhmbo Eck et l.: Impct of Meuemet Eo o Etmto of Pmete of Fte ulto w Le Ted de Stemtc Smplg [7] Coo, G., Tembl, M., Mohe, D., d Gove, B. (007). A compo of Dect v Self-epoted Meue fo Aeg Heght, Weght d Bod M Idex: temtc evew. Obet Rev. 8: [8] Plke, M., Steve, J., Flegl, K., d Rut, P. (997). Pedcto equto do ot elmte temtc eo elf-epoted bod m dex. Obet Reech, 5(4): [9] Stommel, M. Schoebo, C. (009). Accuc d uefule of BMI meue bed o elf-epoted weght d heght: fdg fom e he d h 00006, BMC Publc Hel, 9(4). [0] Belloc, N. (954). Vldto of mobdt uve dt b compo w hoptl ecod, Joul of Amec Stttc Aocto, 49: [] G, P. (955). The Memo Fcto Socl Suve. Joul of Amec Stttc Aocto, 50: [] Sge, O. K, D. R d Smmo, W (959). Hel Stttc fom ecod ouce d houehold tevew comped. Poceedg of e ocl Stttc electo of Amec Stttc Aocto, pge 6-5. [3] Tuell, R d Elo, J. (959). Choc Ille Lge Ct. Hvd vet Pe Cmbdge M. [4] Sädl, C. (99). Model ted Suve Smplg. Spge-velg New Yok, Ic, SA. [5] Nbwg, R. (00). Effect of meuemet eo o populto dom ode whe mplg temtcll. publhed poect deptmet of Memtc Kett vet. [6] Roell, L. C. Coe, P. Stukel, T. Mutd, C. Hux, J. d Muel, D. G. (0). The fluece of meuemet eo o clbto, dcmto d ovell etmto of k pedcto model. ulto Hel Met; 0:0. do:0:86/ [PMC ]. [7] O Nel, D. d Olve, S. (03). The coequece of meumet eo whe etmtg e mpct of obet o come. IZA Joul of Lbo Ecoomc, (3). [8] Subm, J. d Sgh, S. (04). Etmto of populto me e peece of le ted. Commucto e Stttc-Theo d Meod, 43. [9] Ouko, A., Cheuot, W., d Eml, K. (04). Effect of meuemet eo o populto etmte fom mple geeted fom ttfed populto ough temtc mplg techque. Expet Joul of Ecoomc, :0-3. [0] Gellet, E. Golde, M. H (06). The effect of Rdom eo o dgo ccuc llutted w e opometc dgo of mlutto. PLoS ONE (): e do [] Mukhopdh, P. (998). Theo d Meod of Suve Smplg. Petce-Hll of Id Pvte Ltd. [] Hovtz, D. d Thompo, D. (95). A geelzto of Smplg Wout Replcemet Fom Fte vee. Joul of Amec Stttc Aocto, 47: [3] Dog, S. d Chudh, F. S (986). Theo d Al of Smple Suve Deg. New Age Ietol (P) Ltd. [4] Wolte, K. (985). Itoducto to Vce Etmto, Spge-Velg, New Yok. [5] Coch, W. G. (977). Smplg Techque, Joh Wle d So, d edto. [6] Yte, F. (948). Stemtc mplg, Phloophcl Tcto of e Rol ocet of Lodo, A4: