Optimal Allocation in Domains Mean Estimation Using Double Sampling with Non-Linear Cost Function in the Presence of Non-Response

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1 America Joural of Theoretical a Applie tatistics 8; 7(: oi:.648j.ajtas.87. I: (Prit; I: (Olie Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose Alilah Dai Aekeya,, Ouma Christopher Oyago, yogesa Keey Departmet of Mathematics, Masie Muliro Uiersity of ciece a Techology, Kakamega, Keya Departmets of tatistics a Actuarial ciece, Keyatta Uiersity, airobi, Keya aress: Correspoig author To cite this article: Alilah Dai Aekeya, Ouma Christopher Oyago, yogesa Keey. Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose. America Joural of Theoretical a Applie tatistics. Vol. 7, o., 8, pp oi:.648j.ajtas.87. Receie: December 3, 7; Accepte: Jauary 5, 8; Publishe: February, 8 Abstract: tuies hae bee carrie out o omai mea estimatio usig o-liear cost fuctio. Howeer little has bee oe o omai stratum estimatio usig o-liear cost fuctio usig ratio estimatio i the presece of o-respose. This stuy eelops a metho of optimal stratum sample size allocatio i omai mea estimatio usig ouble samplig with o-liear cost fuctio i the presece of o- respose. To obtai a optimum sample size, Lagragia multiplier techique is employe by miimizig precisio at a specifie cost. I the estimatio of the omai mea, auiliary ariable iformatio i which the stuy a auiliary ariables both suffers from o-respose i the seco phase samplig is use. The epressios of the biases a mea square errors of propose estimator has also bee obtaie. Keywors: Optimal Allocatio, Double amplig, o-liear Cost Fuctio, o-respose. Itrouctio.. Domais I samplig, estimates are mae i each of the class ito which the populatio is subiie. uch subgroups or classes are kow as the omai of stuy. Uits of omais may sometimes be ietifie prior to samplig. uch omais are calle plae omais. For uplae omai the uits caot be ietifie prior to samplig a hece the estimates of certai omais is ofte eiet oly after the samplig esig has bee ietifie or after the amplig a fiel work hae bee complete. Hece the size of uplae omai caot be cotrolle. The sample sizes for sub-populatios are raom ariables sice formatio of these sub-populatios is urelate to samplig. Accorig to Eurostat [4] the precisio threshol a or miimum effectie sample sizes are set up for effectie plae omais. The miimum sample sizes require to achiee a relatie margi error of.k% for the total Y (Domai total of a stuy ariable y oer omai U of size gie by Where Z α Z. α y α y (mi K Y Z ( y is the ariace of y oer the omai a is the percetile alue at ( α % of ormal istributio with mea a ariace, K is the relatie margi of error epresse as a proportio while.k% is the relatie margi error epresse as a percetage. The populatio alue Y a y is ukows a hae to be estimate usig ata from auiliary sources... Optimal Allocatio with o-liear Cost Fuctio Optimal sample allocatio ioles etermiig the

2 46 Alilah Dai Aekeya et al.: Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose sample size,,..., H that miimizes the arious cost characters uer a gie samplig buget C (where C is the upper limit of the total cost of the surey. Liear cost fuctio is appropriate whe the cost iole is associate with o-trael actiities of surey e.g rawig sample, preparig surey methos, locatig, ietifyig, iteriewig respoets a coig ata. uch a liear cost fuctio ca be of the form, C c c H h h h where c cost of classificatio per uit size of the first sample c h Cost of measurig a uit i stratum h h umber of uits i stratumh. Geerally the aboe liear cost fuctio is mostly applicable whe the major cost item is that of takig the measuremets o each uit without cosierig the cost of the istace betwee the sample uits. Cheryak [] propose optimum allocatio i ouble samplig for stratificatio with a o-liear cost fuctio. The propose o-liear cost fuctio is of the form, ( α L C c c, α > where, c is the cost of k k k classificatio per uit a c k is the cost of measurig a uit i stratumk. Aother o-liear cost fuctio propose was logarithmic i ature of the form, L log k k k k C c c W Okafor a Lee [9] employe the ouble samplig metho to estimate the mea of the auiliary ariable a proceee to estimate the mea of the stuy ariable i a similar way as Cochra [3]. I this metho ouble samplig ratio a regressio estimatio was cosiere. The istributio of the auiliary iformatio was ot kow a hece the first phase sample was use to estimate the populatio istributio of the auiliary ariable while the seco phase was use to obtai the require iformatio o the ariable of the iterest. The optimum samplig fractio was erie for the estimators at a fie cost. Performace of the propose estimators was compare with those of Hase a Hurwitz [5] estimators without cosierig the cost. It was ote that for the results for which cost compoet was ot cosiere, regressio estimator fuctios were cosistet tha the Hase a Hurwitz [5] estimator. Tschuprow [] a eyma [8] propose the allocatio proceure that miimizes ariace of sample mea uer a liear cost fuctio of sample size H h h, where h is the size of the stratum. eyma [8] use Lagrage multiplier optimizatio techique to get optimum sample sizes for a sigle ariable uer stuy. Holmberg [6] aresse the problem of compromise allocatio i multiariate tratifie samplig by takig ito cosieratio miimizatio of some of the ariaces or coefficiet of ariatio of the populatio parameters a of some of the efficiecy losses which may be as a result of icrease i the ariace ue to the use of compromise allocatio. aii [] eelope a metho of optimum allocatio for multiariate stratifie two stage samplig esig by usig ouble samplig. I this metho the problem of etermiig optimum allocatios was formulate as o-liear programmig problem (LPP i which each LPP has a coe objectie fuctio uer a sigle liear costraits. The Lagrage multiplier techique was use to sole the formulate LPPs. Kha et al. [7] propose a quaratic cost fuctio for allocatig sample size i multiariate stratifie raom samplig i the presece of o-respose i which a separate liear regressio estimator is use. I this multi-objectie o-liear iteger programmig problem, a etee leicographic goal programmig was use for solutio purpose a compariso mae with iiiual optimum techiques. It is obsere that i the allocatio techiques, the etee leicographic goal programmig gies miimum alues of coefficiet of ariatio tha the iiiual optimum a goal programmig techique. Chouhry [] cosiere sample allocatio issues i the cotet of estimatig ub-populatios (stratum a omai meas as well as the aggregate populatio meas uer stratifie simple raom samplig. I this metho o-liear programmig was use to obtai the optimal sample allocatio to the strata that miimizes the total sample sizes subject to a specifie tolerace o the coefficiet of ariatio of the estimators of strata a populatio meas. From the preious stuies, a umber of researchers hae cosiere a liear cost fuctio whe estimatig omais. I ealig with o-respose most of them hae cosiere subsamplig while holig to the iea that the respose mechaism is etermiistic. This paper therefore focuses o the estimatio of omai mea usig ouble samplig for ratio estimatio with o-liear cost fuctio with a raom respose mechaism. I this stuy we therefore establish a efficiet a cost effectie metho of estimatig omais whe the trael compoet is iclusie a it is ot liear.. Estimatio of Domai Mea a Variace i the Presece of o- Respose.. Itrouctio The problem of o-respose is iheret i may sureys. It always persists ee after call-backs. The estimates obtaie from icomplete ata will be biase especially whe the respoets are ifferet from the o-respoets. The orespose error is ot so importat if the characteristics of the o-respoig uits are similar to those of the respoig uits. Howeer, such similarity of characteristics betwee two types of uits (respoig a o-respoig is ot always attaiable i practice. I ouble samplig whe the problem of

3 America Joural of Theoretical a Applie tatistics 8; 7(: o-respose is preset, the strata are irtually iie ito two isjoit a ehaustie groups of respoets a orespoets. A sub-sample from o-respoig group is the selecte a a seco more etesie attempt is mae to the group so as to obtai the require iformatio. Hase a Hurwitz [5] propose a techique of ajustig the orespose to aress the problem of bias. The techique cosists of selectig a sub-sample of the o-respoets through specialize efforts so as to obtai a estimate of orespoig uits i the populatio. This sub-samplig proceure albeit costly, it s free from ay assumptio hece, oe oes ot hae to go for a hure percet respose which ca be substatially more epesie. I eelopig the cocept of omai theory with orespose the followig assumptios are mae; i. Both the omai stuy a auiliary ariables suffers from o-respose. ii. The respoig a o-respoig uits are the same for the stuy a auiliary characters. iii. The iformatio o the omai auiliary ariable X is ot kow a hece X is ot aailable. i. The omai auiliary ariables o ot suffer from orespose i the first phase samplig but suffers from o-respose i the seco phase of samplig... Propose Domai Estimators Let U be a fiite populatio with kow first stage uits. The fiite populatio is iie ito D omais; U, U,..., U D of sizes,,...,,..., D respectiely. Further, let U be the omai costituets of ay populatio size which is assume to be large a kow. LetU a be efie as, U U a respectiely. D D Let Y a X be the omai stuy a auiliary ariables respectiely. Further, let Y a X be their respectie omai populatio meas a auiliary meas with y i,,3,..., i,,..., obseratios o i ( a ( i the i th uit. I estimatig the omai auiliary populatio mea X ouble samplig esig is use. A large first phase sample of size is selecte from uits of the populatio by simple raom samplig without replacemet (RWOR esig from which out first sample uits fallig i the th omai. The assumptio here is that all the uits supply iformatio of the auiliary ariable X at first phase. A smaller seco phase sample of size is selecte from by RWOR from which out of seco phase sample uits fall i the th omai. For estimatig the omai populatio mea X of the auiliary ariables X from a large first phase sample of size, alues of the obseratios i ( i,,3,..., are obtaie a a sample auiliary omai mea is compute. From the seco sample of size, let y a i i be the omai stuy a auiliary obseratios with ( i,,3,...,. Let uits supply the iformatio o i y i a respoets while i be the o-respoets for both the stuy a the auiliary omai ariables respectiely such that,. For the o-respoet group at the seco phase samplig, a RWOR of r uits is selecte with a ierse samplig rate of such that, r, With > All the r uits respo after makig etra efforts of subsamplig o-respoig uits. I eelopig the framework of ouble samplig there are two strata that are o-oerlappig a isjoit. tratum oe cosist of those uits that will respo i the first attempt of the seco phase populatio mae up of uits a stratum two cosist of those uits that woul ot respo i the first attempt of phase two with omai populatio uits. Both a uits are ot kow i aace. The stratum weights of the respoig a o- respoig groups are efie by W a with their estimators efie by Wˆ w respectiely. W respectiely Wˆ w a Followig the Hase a Hurwitz [5] techiques, the ubiase estimator for estimatig the omai populatio mea usig ( r obseratios o character is gie by; y y y r w y r y i omai stuy w y ( imilarly the estimate for omai auiliary ariable is gie by; r w w (3 r Where y a are the sample omai meas for the

4 48 Alilah Dai Aekeya et al.: Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose obseratio y i a i respectiely. I estimatig the oerall omai populatio mea i the presece of o-respose, ouble samplig ratio estimatio of the omai mea is use. Defie; ˆ y ˆ y Y r a Y. r. R With the assumptio that, E E X, E ( y 3. Mea quare Error of the Ratio Estimator Y (4 The epressio for the Mea square error (ME of Y R a Y R are erie by the use of the Taylor's series approimatio. Let ε ε ε y Y Y ( ε y Y X ( ε X X X X X ( ε (5 With the assumptio that E ( ε E ( ε E ( ε Further efie; E ( ε y Y E Y ( y Y E Ey Y Y Y Var ( y Y VE E3 ( y EV E3 ( y EEV3 ( y r Y W Y y y y (6 C C W C y y y y Variace of the whole omai populatio mea of the stuy ariable Y y Variace of the omai populatio mea for the stratum of o-respoets for tratum of o-respoets for the stuy ariable Y Cosier also X X X Eε E E X Var ( X V E ( E V ( r X W X X Variace of the whole omai populatio mea of the auiliary ariable X The ierse samplig rate Variace of the omai populatio mea for the stratum of o-respoets for tratum of o-respoets for the auiliary ariable X (7

5 America Joural of Theoretical a Applie tatistics 8; 7(: et cosier X X X ( ε E E E X Var ( X Cosier, X X (8 E y Y X ε ε E Y X E( y Y ( X Co( y CoE ( y E ( E Co( y ECo( y r E Co y E Co y r ( ( y y ρ y ρ y W (9 et, y Y X Eε ε Y X E( y Y ( X Co( y ( ( ( ( CoE y, E ECo y E Co yr r y ρ y Cosier, X X Eε ε E X X ( ( X E E X X X X E ( X 3.. Mea quare Error (ME of the Ratio Estimator Y ˆ R a Y ˆ with the ample ize Allocatio R The ratio estimator of R ( ( ˆ ˆ Y a Y ca be efie as; R

6 5 Alilah Dai Aekeya et al.: Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose ˆ y ˆ y Y r a Y. r respectiely. R Propositio The mea square error (ME of the estimator efie by ˆ y Y. R r is gie by; y W R R R y ρ y y R R R y ρ y y R R With the otatios efie as: y Variace of the whole omai populatio mea of the stuy ariable Y y Variace of the omai populatio mea for the stratum of o-respoets for tratum of o-respoets for the stuy ariable Y Variace of the whole omai populatio mea of the auiliary ariable X The ierse samplig rate Variace of the omai populatio mea for the stratum of o-respoets for tratum of o-respoets for the auiliary ariable X R Populatio ratio of Y to X Proof By efiitio, ˆ ( R ˆ ME Y E Y Y R y E. Y ubstitutig the alues of equatios (5 we obtai ( ε X ( ε X ( ε Y E Y ( ε ( ε ( ε Y E ( (... Y E ε ε ε ε ε ε ε Y E ( ε ε ε Y E ( ε E ( ε E ( ε E ( ε ε E ( ε ε E ( ε ε Y Y Y Y y y y W X X W X y ρ y y ρ y ρ W y y X ] y y R

7 America Joural of Theoretical a Applie tatistics 8; 7(: R W R ρ R y y y ρ R y y ρ R y y { ρ } y y R y R y W R ρ R y y y y R W R Propositio The mea square error (ME of the ratio estimator W R ˆ Y y. is gie by; R y R R y ρ y y R R With the otatios as efie i propositio aboe Proof ME of y ρ y y R R R ˆ ˆ ˆ Y R ME ( Y EY R Y R y E. Y E Y ( ε ( ε ( ε Y E( ε ε ε ε ε ( ε ε... Y E ( ε ε ε Y E ( ε E ( ε E ( ε E ( ε ε E ( ε ε ( ε ε Y Y Y Y y y y W

8 5 Alilah Dai Aekeya et al.: Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose W X X X y y ρ y W ρ y y ρ y X y y W y R R W R ρ R W ρ R y y y y ρ y R y R ( ρ y y R y R y W ( y R ρ y R y y R W R 3.. Optimal Allocatio i Double amplig for Domai Estimatio A optimum size of a sample is require so as to balace the precisio a cost iole i the surey. The optimum allocatio of a sample size is attaie either by miimizig the precisio agaist a gie cost or miimizig cost agaist a gie precisio. I this stuy, a o-liear cost fuctio has bee cosiere. Deote the cost fuctio for the ratio estimatio by size ( C c c c c r ( c The cost of measurig a uit i the first sample of c The cost of measurig a uit of the first attempt o y with seco phase sample size. c The uit cost for processig the respoe ata of y at the first attempt of size. c The uit cost associate with the sub-sample of size r from o-respoets of size Howeer the first sample of size a sub-samples of size r are ot kow util the first attempt is carrie out. The cost will therefore be use i the plaig for the surey. Hece the epecte cost alues of sizes a r will be gie by; W a r W.. Hece the epecte cost fuctio is; C c ( c c c W E C

9 America Joural of Theoretical a Applie tatistics 8; 7(: W c c c. W c (3 C ( 3.3. Results of Double amplig for Domai Estimatio i the Presece of o-respose Propositio 3 The ariace for the estimate omai mea for the ˆ y estimate omai mea Y. R is miimum for a specifie cost C whe, ω c ( W R R c ( c c W ( W R R. R ( c c W y R ω > R y ± ρ y y R R R y ± ρ y y R R Proof To etermie the optimum alues of, a miimizes ariace at a fie cost, efie G ( W W that y R R W c ( c c W c. C (4 To obtai the ormal equatios, the epressio of Equatio (4 is ifferetiate partially with respect to, a, a the partial eriaties are equate to zero ( G W y ( R c y R ( y R c ( R y c y R ( Let ω >, thus, ( c ω c ω ω c c et the partial eriatie with respect to ( W R G W c W c W W R W R c W Cosier the equatio G ( W c obtaie as; (5 R y W R R W c ( c c W c. C (6 a But from (5, c R R (7 c ubstitutig this i Equatios (7 we obtai ito (6 we obtai

10 54 Alilah Dai Aekeya et al.: Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose G ( W c y W R R R ( R c c c W c W C c (8 The partial eriatie of the equatio (8 with respect to is obtaie as ( G W W ( c c W R R ( W c c W R R ( c c W W R R W R R ( c c W W R R c c W Where Thus, But c. from equatio (7 c R ( W R R. R ( c c W To obtai the alues of, a are substitute i the cost fuctio equatio (3 a the sole for the alue of. uppose the cost fuctio is gie by The, W c c c. W c C ( C W R R W R ω c W R c c c W c c c W c c c W W R ω R ( R c c c W C c W c c W (9 Let, A ω ( c ( W R R B c c W W c R c c W C C The equatio (9 becomes; A B C ( If a substitutig this alue i the equatio ( we obtai a liear equatio of the form

11 America Joural of Theoretical a Applie tatistics 8; 7(: With the alues of A a B efie as; A B C A ω ( c B c c W W c W R R R c c W, ( a C C olig the liear equatio solutio obtaie is, A B C Whe a substitutig this alue i the equatio 3 ( we obtai a liear equatio of the form, 4 A B C ( With the alues of A efie as; A ω ( c While B a C remais as earlier efie olig the equatio ( solutio obtaie is, A ( B 4AC B Propositio 4 If the epecte cost fuctio is of the form C W c log c c. W c the the ariace of the 4 estimate omai mea y is miimum for a specifie cost C if; Proof The proof for ω c ( W R R c ( c c W ( W R R. R ( c c W a propositio 3 aboe. For techique is use. Let, is the same as the oe i the Lagragia multiplier G ( W W y R R W c c c W c C log. ( To obtai the ormal equatios for the epressio ( the equatio is ifferetiate partially with respect to partial eriaties are equate to zero ( y R c G W y R c y R c a the

12 56 Alilah Dai Aekeya et al.: Optimal Allocatio i Domais Mea Estimatio Usig Double amplig with o-liear Cost Fuctio i the Presece of o-respose y c R But, y R ω >, thus, ω ω c c To sole for, let the ariace be gie as V the substitute the alues of, a ubstitute the alues of, a G ( W y W R V R ito the equatio, y W R V R y ( y ( R W R R W V (3 R ito the equatio (3 a simplify to obtai, ( ( c W R c R c W W c R V (4 Let, V y V, A c R R R B W c c W W c a ( ( Thus equatio (4 becomes, A B V (5 olig for i equatio ( the solutio becomes, 4. Coclusio ( 4 B AC B A It is ote that alue of ierse samplig rate ( ot epe o the alue of the Lagragia multiplier (. Further the alue of samplig rate, <, if c (the uit o cost associate with the sub-sample of size r from o- respoets of size is less tha both c (the uit cost for processig the respoe ata of y at the first attempt of size a c (the cost of measurig a uit of the first attempt o y with seco phase sample size a also whe the alue of R is ot too large relatie to R. The seco phase sample size ( will be miimum if the alue of ( W R R > but less tha. If the alue of ω y < a R y > R with the alues of c (the cost of measurig a uit i the first sample of size ot beig too large to the relatie ω the the alue of (size of the first sample will be miimum. These miimum alues therefore make the theoretic cost surey of the propose estimator as miimal as possible.

13 America Joural of Theoretical a Applie tatistics 8; 7(: Refereces [] Cheriyak O. I., (. Optimal allocatio i stratifie samplig a ouble samplig with o- liear cost fuctio, Joural of Mathematical cieces 3, 4 pp [] Chouhry H. G., Rao, J.. K, a Michael A., Hiiroglou, (. O sample allocatio for efficiet omai estimatio, urey methoology, 38 ( pp [3] Cochra W. G., (977 amplig techiques. ew York: Joh Wiley a os, (977. [4] Eurostat., (8. Itrouctio to ample Desig a Estimatio Techiques, urey amplig Referece Guielies. Luembourg; Office for Publicatio of the Europea Commuities pp. 36. [5] Hase M. H. a Hurwitz W. W, (946. The problem of o-respose i sample sureys. The Joural of the America tatistical Associatio, [6] Holmberg A., (. A multi-parameter perspectie o the choice of samplig esigs i sureys. Joural of statistics i trasitio. 5 (6 pp [7] Kha. U., Muhamma Y.., a Afga., (9. Multiobjectie compromise allocatio stratifie samplig i the presece of o-respose usig quaratic cost fuctio. Iteratioal Joural of Busiess a social sciece. 5 (3. pp [8] eyma, C. a Jerzy D. (934.; O the Two Differet Aspects of the Represetatie methos of stratifie samplig a the metho of purposie selectio, Joural of royal statistical society. 97 (4 pp [9] Okafor F. C, (. Treatmet of o-respose i successie samplig, tatistica, 6 ( [] aii M., a Kumar A, (5. Metho of Optimum allocatio for Multiariate tratifie two stage amplig esig Usig ouble amplig. Joural of Probability a tatistics forum, 8, pp [] Tschuprow a Al A., (93. O mathematical epectatio of the momets of frequecy istributio i the case of correlate obseratio (chapters 4-6 Metro ( pp , (93.