SOME EFFECTIVE ESTIMATION PROCEDURES UNDER NON-RESPONSE IN TWO-PHASE SUCCESSIVE SAMPLING

Singh, G. N.; Khe an, M.; Mau ya, S. A icle SOME EFFECTIVE ESTIMATION PROCEDURES UNDER NON- RESPONSE IN TWO-PHASE SUCCESSIVE SAMPLING S a is ics in T ansi ion New Se ies P o ided in Coope a ion wi h: Polish S a is ical Associa ion Sugges ed Ci a ion: Singh, G. N.; Khe an, M.; Mau ya, S. (2016) : SOME EFFECTIVE ESTIMATION PROCEDURES UNDER NON-RESPONSE IN TWO-PHASE SUCCESSIVE SAMPLING, S a is ics in T ansi ion New Se ies, ISSN 2450-0291, Exeley, New Yo k, NY, Vol. 17, Iss. 2, pp. 163-182, h ps://doi.o g/10.21307/s a ans-2016-012 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/207805 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ STATISTICS IN TRANSITION new se ies, June 2016 163 STATISTICS IN TRANSITION new se ies, June 2016 Vol. 17, No. 2, pp. 163–182 SOME EFFECTIVE ESTIMATION PROCEDURES UNDER NON-RESPONSE IN TWO-PHASE SUCCESSIVE SAMPLING G. N. Singh 1 , M. Khe an 2 , S. Mau ya 3 ABSTRACT This wo k is designed o assess he e ec o non- esponse in es ima ion o he cu en popula ion mean in wo-phase successi e sampling on wo occasions. Sub-sampling echnique o non- esponden s has been used and exponen ial me hods o es ima ion unde wo-phase successi e sampling a angemen ha e been p oposed. P ope ies o he p oposed es ima ion p ocedu es ha e been examined. Empi ical s udies a e ca ied ou o jus i y he sugges ed es ima ion p ocedu es and sui able ecommenda ions ha e been made o he su ey p ac i ione s. Key wo ds: non- esponse, successi e sampling, wo-phase sampling, mean squa e e o , op imum eplacemen s a egy. 1. In oduc ion In collec ing in o ma ion h ough sample su eys, he e may a ise nume ous p oblems; one o hem is non- esponse. I equen ly occu s in mail su eys, whe e some o he selec ed uni s may e use o e u n back he illed in ques ionnai es. An es ima e ob ained om such an incomple e su ey may be misleading, especially when he esponden s di e signi ican ly om he non- esponden s, because he es ima e may be a biased one. Hansen and Hu wi z (1946) sugges ed a echnique o sub-sampling o non- esponden s o handle he p oblem o non- esponse. Coch an (1977) and Fabian and Hyunshik (2000) ex ended he Hansen and Hu wi z (1946) echnique o he si ua ion when besides he in o ma ion on he cha ac e unde s udy, in o ma ion on auxilia y cha ac e is also a ailable. Recen ly, Choudha y e al. (2004) Singh and P iyanka (2007), Singh and Kuma 1 Depa men o Applied Ma hema ics, Indian School o Mines, Dhanbad 826004. E-mail: [email protected]. 2 Depa men o Applied Ma hema ics, Indian School o Mines, Dhanbad 826004. E-mail: muk i.khe [email protected]. 3 Depa men o Applied Ma hema ics, Indian School o Mines, Dhanbad 826004. 164 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion .... (2009, 2010), Singh e al. (2011) and Ga cia Luengo (2013) used he Hansen and Hu wi z (1946) echnique o he es ima ion o popula ion mean on he cu en occasion in wo-occasion successi e sampling. I he s udy cha ac e o a ini e popula ion is subjec o change o e ime, a single occasion su ey is insu icien . Fo such a si ua ion successi e sampling p o ides a s ong ool o gene a ing eliable es ima es o e di e en occasions. Sampling on successi e occasions was i s conside ed by Jessen (1942) in he analysis o a m da a. The heo y o successi e ( o a ion) sampling was u he ex ended by Pa e son (1950), Eckle (1955), Rao and G aham (1964), Sen (1971, 1972, 1973), Gup a (1979), Das (1982) and Singh and Singh (2001) among o he s. In sample su eys, he use o auxilia y in o ma ion has shown i s signi icance in imp o ing he p ecision o es ima es o unknown popula ion pa ame e s. When he popula ion pa ame e s o auxilia y a iable a e unknown be o e s a o he su ey we go o wo-phase (double) sampling s uc u e o p o ide he eliable es ima es o he unknown popula ion pa ame e s. Singh and Singh (1965) used wo-phase (double) sampling o s a i ica ion on successi e occasions. Recen ly, Singh and P asad (2011) and Singh and Homa (2014) applied wo-phase sampling scheme wi h success in he es ima ion o he cu en popula ion mean in wo- occasion successi e sampling. The aim o he p esen wo k is o s udy he e ec o non- esponse when i occu s on a ious occasions in wo-occasion successi e ( o a ion) sampling. Recen ly, Bahl and Tu eja (1991), Singh and Vishwaka ma (2007) and Singh and Homa (2013) sugges ed exponen ial ype es ima o s o popula ion mean unde di e en ealis ic si ua ions. Mo i a ed wi h he domina ing na u e o hese es ima o s and u ilizing he in o ma ion on a s able auxilia y a iable wi h unknown popula ion mean o e bo h occasions, some new exponen ial me hods o es ima ion ha e been p oposed o es ima e he cu en popula ion mean in wo- phase successi e ( o a ion) sampling a angemen .. The Hansen and Hu wi z (1946) echnique o sub-sampling o non- esponden s has been used o educe he nega i e e ec s o non- esponse. P ope ies o he p oposed es ima o s a e examined and hei empi ical compa isons a e made wi h he simila es ima o and wi h he na u al successi e sampling es ima o when comple e esponse is obse ed on bo h occasions. Resul s a e in e p e ed and ollowed by sui able ecommenda ions. 2. Sample s uc u es and symbols Le U = (U1, U2, -, -, -, UN) be he ini e popula ion o N uni s, which has been sampled o e wo occasions. The cha ac e unde s udy is deno ed by x(y) on he i s (second) occasion espec i ely. I is assumed ha he non- esponse occu s only in s udy a iable x(y) and in o ma ion on an auxilia y a iable z (s able o e occasion), whose popula ion mean is unknown on bo h occasions, is a ailable and posi i ely co ela ed wi h s udy a iable. Since we ha e assumed ha non- STATISTICS IN TRANSITION new se ies, June 2016 165 esponse occu s on bo h occasions, he popula ion can be di ided in o wo classes – hose who will espond a he i s a emp and hose who will no on bo h occasions. Le he sizes o hese wo classes be * 1 N and * 2 N espec i ely on he i s occasion and he co esponding sizes on he cu en (second) occasion be N1 and N2, espec i ely. To u nish a good es ima e o he popula ion mean o he auxilia y a iable z on he i s occasion, a p elimina y sample o size n' is d awn om he popula ion by he simple andom sampling wi hou eplacemen (SRSWOR) me hod, and in o ma ion on z is collec ed. Fu he , a second-phase sample o size n ( n' > n) is d awn om he i s -phase (p elimina y) sample by he SRSWOR me hod and hence o h he in o ma ion on he s udy cha ac e x is ga he ed. We assume ha ou o selec ed n uni s, n1 uni s espond and n2 uni do no espond. Le n2h deno e he size o sub-sample d awn om he non- esponding uni s in he sample on i s occasion. A andom sub-sample sm o m = n  uni s is e ained (ma ched) om he esponding uni s on he i s occasion o i s use on he second occasion unde he assump ion ha hese uni s will gi e comple e esponse on he second occasion as well. Once again, o u nish a esh es ima e o he popula ion mean o he auxilia y a iable z on he second occasion, a p elimina y ( i s -phase) sample o size u' is d awn om he non-sampled uni s o he popula ion by he SRSWOR me hod and in o ma ion on z is collec ed. A second-phase sample o size u = (n-m) = nμ ( u' > u) is d awn om he i s - phase (p elimina y) sample by he SRSWOR me hod and he in o ma ion on s udy a iable y is ga he ed. I is ob ious ha he sample size on he second occasion is also n. He e λ and μ (λ+μ =1) a e he ac ions o he ma ched and esh samples, espec i ely, on he second (cu en ) occasion. We assume ha in he unma ched po ion o he sample on he cu en (second) occasion u1 uni s espond and u2 uni s do no espond. Le u2h deno e he size o he sub-sample d awn om he non- esponding uni s in he esh sample (su) on he cu en (second) occasion. Hence, onwa ds, we use he ollowing no a ions: X, Y, Z : The popula ion means o he a iables x, y and z espec i ely. 1 2 h 1 2h m u u u n n n m m u y , y , y , y , x , x , x , x , z ,z : The sample means o he espec i e a iables based on he sample sizes shown in su ices. '' nu z , z : The sample means o he auxilia y a iable z and based on he i s - phase samples o sizes u' and n' espec i ely. yx xz yz ρ , ρ , ρ : The popula ion co ela ion coe icien s be ween he a iables shown in su ices. 2 2 2 x y z S , S , S : The popula ion a iances o he a iables x, y and z espec i ely. 22 2x 2y S ,S : The popula ion a iances o he a iables x and y espec i ely in he non- esponding uni s o he popula ion. x y z C , C , C : The coe icien s o a ia ion o he a iables x, y and z espec i ely. 166 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion .... 2x 2y C , C : The coe icien s o a ia ion o he a iables x and y in he non- esponding uni s o he popula ion. * *2 N W=N : The p opo ion o non- esponding uni s in he popula ion a i s occasion. 2 N W= N : The p opo ion o non- esponding uni s in he popula ion on he cu en (second) occasion. 22 12 2h 2h nu = and = nu . 3. Fo mula ion o es ima ion s a egy To es ima e he popula ion mean Y on he cu en (second) occasion, wo di e en es ima o s a e conside ed – one es ima o u T based on sample su o size u d awn a esh on he second occasion and he second es ima o m T based on he sample sm o size m, which is common o bo h occasions. Since he non- esponse occu s in he samples sn and su, we ha e used he Hansen and Hu wi z (1946) echnique o p opose he es ima o s Tu and Tm. Hence, he es ima o s Tu and Tm o es ima ing he cu en popula ion mean Y a e o mula ed as '*' *uu n m n u u m m '* u u n m m z -z x -x z T = y exp and T = y exp z +z x +x z             whe e 1 2h 1 2h 1 n 2 n 1 u 2 u ** nu n x +n x u y +u y x = and y = . nu Combining he es ima o s Tu and Tm, inally we ha e he ollowing es ima o o popula ion mean Y on he cu en (second) occasion   u m T = φ T + 1-φ T (3.1) whe e φ (0 φ 1) is he unknown cons an (scala ) o be de e mined unde ce ain c i e ions. 4. P ope ies o he es ima o T Since he es ima o s u m T and T a e exponen ial ype es ima o s, he popula ion mean Y a e biased, he e o e he esul ing es ima o T de ined in STATISTICS IN TRANSITION new se ies, June 2016 167 equa ion (3.1) is also a biased es ima o o Y . The bias B (.) and he mean squa e e o M (.) o he es ima o T a e de i ed up o he i s o de o app oxima ions using he ollowing ans o ma ions: * m 1 u 2 u 3 m 4 n 5 ' * ' n 6 n 7 m 8 u 9 u 10 y (1 e )Y, y (1 e )Y, y (1 e )Y, x (1 e )X, x (1 e )X, x (1 e )X, x (1 e )X, z (1 e )Z, z (1 e )Z, z (1 e )Z,                     ' n 11 z (1 e )Z, such ha i E(e ) = 0 , i e <1 i = 1, 2, 3, - - -, 11. Unde he abo e ans o ma ions, he es ima o s u m T and T ake he ollowing o ms: -1 u 3 10 9 10 9 11 T =Y(1+e )exp (e -e ) 1+ (e +e ) 22        (4.1) and -1 -1 m 1 11 8 7 4 7 4 11 T =Y(1+e )(1+e )(1+e ) exp (e -e ) 1+ (e +e ) 22        (4.2) Thus, we ha e he ollowing heo ems: Theo em 4.1. Bias o he es ima o T o he i s o de o app oxima ions is ob ained as u m B(T) = φB(T )+(1-φ)B(T ) (4.3) whe e 2 u z yz y z ' 1 1 3 1 B(T ) = Y - C - ρ C C u u 8 2          and     2 x xz x z xy y x m * 2 2 12x z yz y z ' 1 1 3 1 1 - C + ρ C C - ρ C C m n 8 2 2 B T =Y ( -1) 1 1 1 - W C + - C -ρ C C 8 n m n                    P oo The bias o he es ima o T is gi en by um B(T) = E T-Y = φE(T -Y) + (1-φ) E(T -Y)     um = φB(T )+ 1-φ B(T ) (4.4) whe e u u m m B(T ) = E T -Y and B(T ) = E T -Y .         168 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion .... Subs i u ing he exp essions o Tu, and Tm om equa ions (4.1) and (4.2) in equa ion (4.4), expanding he e ms binomially and exponen ially, aking expec a ions and e aining he e ms up o he i s o de o sample sizes, we ha e he exp essions o he bias o he es ima o T as desc ibed in equa ion (4.3). Theo em 4.2. The mean squa e e o o he es ima o T o he i s o de o app oxima ions is ob ained as         2 2um M( T) = φ M T + 1-φ M T +2φ 1-φ C (4.5) whe e   22 2 u u yz y ' W( -1) 1 1 1 1 1 M(T ) = E T -Y = - -ρ + - + S u u 4 u N u                   (4.6)     xz yx 22 m m y * 1 yz ' 1 1 1 1 1 - +ρ -ρ + - m n 4 m N M(T ) = E T -Y = S ( -1) 1 1 1 + - 1 -2ρ + W m n 4 n                                (4.7) and    2 y um S C = E T -Y T -Y = - N   . (4.8) P oo I is ob ious ha he mean squa e e o o he es ima o T is gi en by       2 2 um M(T) = E T-Y = E φ T -Y + 1-φ T -Y                22 2 2u m u m = φ E T -Y + 1-φ E T -Y +2φ 1-φ E T -Y T -Y         2 2um = φ M(T )+ 1-φ M T +2φ 1-φ C (4.9) Subs i u ing he exp essions o Tu, and Tm om equa ions (4.1)-(4.2) in equa ion (4.9), expanding he e ms binomially and exponen ially, aking expec a ions and e aining he e ms up o he i s o de o sample sizes, we ha e he exp ession o he mean squa e e o o he es ima o T as i is gi en in equa ion (4.5). STATISTICS IN TRANSITION new se ies, June 2016 169 Rema k 4.1. The exp ession o he mean squa e e o in he equa ion (4.5) is de i ed unde he assump ions (i) ha he coe icien s o a ia ion o non- esponse class a e simila o ha o he popula ion, i.e. 2x x 2y y C = C and C = C , and (ii) since x and y a e he same s udy a iable o e wo occasions and z is he auxilia y a iable co ela ed o x and y, looking a he s abili y na u e o he coe icien s o a ia ion, iz. Reddy (1978), he coe icien s o a ia ion o he a iables x, y and z in he popula ion a e conside ed equal, i.e. x y z C = C = C . 5. Minimum mean squa e e o o he es ima o T Since he mean squa e e o o he es ima o T in equa ion (4.5) is he unc ion o unknown cons an φ, i is minimized wi h espec o φ, and subsequen ly he op imum alue o φ is ob ained as m um op M(T )-C φ = M(T )+M(T )-2C . (5.1) Now, subs i u ing he alue o op φ in equa ion (4.5), we ge he op imum mean squa e e o o T as 2 um op um M(T ).M(T )-C M(T) = . M(T )+M(T )-2C (5.2) Fu he , subs i u ing he alues om equa ions (4.6)-(4.8) in equa ion (5.2), we ge he simpli ied alue o M (T)op which is gi en below: 2 2y 3 2 1 op 2 6 5 4 S a+μa +μ a M(T) = a+μa +μ a n . (5.3) whe e 2 2 2 2 1 2 3 4 5 6 * 1 0 0 2 2 1 1 1 * 2 1 1 0 yz 1 xz xy a = ac+k , a = ad+cb-k , a = bd, a = c-a+2k , a = a-b+d-2k , a = b, 1 a = -( + a ), b= a +1+( -1)W, c = d +c + - ( -1)W , 4 1 1 1 d = 1- +(1- )d + ( -1)W , k =-1, a = -ρ , c = +ρ -ρ , 4 4 4 1 yz 22 1 2 1 2 '' 2h 2h d = 1-2ρ, nu n n n = , = , = , = and = . N n u u n 170 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion .... 6. Op imum eplacemen s a egy Since he mean squa e e o o he es ima o T gi en in equa ion (5.3) is he unc ion o μ ( ac ions o he sample o be d awn a esh a he second occasion), he op imum alue o μ is de e mined o es ima e he popula ion mean Y wi h maximum p ecision and lowes cos . To de e mine he op imum alue o μ, we minimized he mean squa e e o o he es ima o T gi en in equa ion (5.3) wi h espec o μ, which esul s in quad a ic equa ion in μ and he espec i e solu ions o μ, say ˆ μ , a e gi en below: 2 1 2 3 pμ +2p μ+p =0 (6.1) 2 2 2 1 3 1 -p ± p -p p ˆ μ = p (6.2) whe e 1 1 5 2 4 2 1 6 3 4 3 2 6 3 5 p = a a -a a , p = a a -a a and p = a a -a a . F om equa ion (6.2) i is ob ious ha eal alues o ˆ μ exis i he quan i ies unde squa e oo a e g ea e han o equal o ze o. Fo any combina ions o co ela ions ρyx, ρxz and ρyz, which sa is y he condi ions o eal solu ions, wo eal alues o ˆ μ a e possible. Hence, while choosing he alues o ˆ μ , i should be emembe ed ha ˆ 0μ1 . I bo h he alues o ˆ μ sa is y he s a ed condi ion, we chose he smalle alue o ˆ μ as i will help in educing he cos o he su ey. All o he alues o μ a e inadmissible. Subs i u ing he admissible alue o ˆ μ , say (0) μ , om equa ion (6.2) in o equa ion (5.3), we ha e he op imum alue o he mean squa e e o o T, which is shown below: 2 (0) (0)2 y 03 2 1 op (0) (0)2 6 5 4 S a+μ a +μ a M(T ) = a+μ a +μ a n . (6.3) 7. Some special cases Case 1: When non- esponse occu s only a i s occasion When non- esponse occu s only a i s occasion, he es ima o o he mean Y on he cu en occasion may be ob ained as   * * * 1u m T = φ ξ + 1-φ T (7.1) STATISTICS IN TRANSITION new se ies, June 2016 177 Table 3. Pe cen age ela i e loss * 1 L in he p ecision o he es ima o T* wi h espec o 1 ξ No e: ‘*’ indica es *(0) μ does no exis . Table 4. Pe cen age ela i e loss * 2 L in he p ecision o he es ima o T* wi h espec o 2 ξ No e: ‘*’ indica es *(0) μ does no exis . W 0.05 0.10 0.15 0.20 ρyx 2 yz *(0) μ * 1 L *(0) μ * 1 L *(0) μ * 1 L *(0) μ * 1 L 0.6 1.5 0.6 0.7 0.8 0.9 * 0.5489 0.3748 0.2828 - 0.1481 0.3426 0.5457 * 0.5667 0.3911 0.2961 - 0.2827 0.6617 1.0599 * 0.5832 0.4065 0.3089 - 0.4057 0.9596 1.5452 * 0.5984 0.4212 0.3212 - 0.5185 1.2385 2.0041 2.0 0.6 0.7 0.8 0.9 * 0.5667 0.3911 0.2961 - 0.2827 0.6617 1.0599 * 0.5984 0.4212 0.3212 - 0.5185 1.2385 2.0041 * 0.6258 0.4484 0.3445 - 0.7180 1.7458 2.8508 * 0.6497 0.4732 0.3663 - 0.8890 2.1954 3.6142 0.8 1.5 0.6 0.7 0.8 0.9 0.6356 0.5141 0.4277 0.3613 0.1025 0.2075 0.3345 0.4977 0.6443 0.5234 0.4368 0.3698 0.1997 0.4057 0.6552 0.9754 0.6525 0.5324 0.4456 0.3780 0.2920 0.5951 0.9628 1.4345 0.6604 0.5411 0.4542 0.3861 0.3798 0.7764 1.2582 1.8759 2.0 0.6 0.7 0.8 0.9 0.6443 0.5234 0.4368 0.3698 0.1997 0.4057 0.6552 0.9754 0.6604 0.5411 0.4542 0.3861 0.3798 0.7764 1.2582 1.8759 0.6752 0.5574 0.4705 0.4016 0.5430 1.1163 1.8150 2.7098 0.6887 0.5727 0.4859 0.4163 0.6916 1.4292 2.3308 3.4842 W 0.05 0.10 0.15 0.20 yx 2 yz *(0) μ * 2 L *(0) μ * 2 L *(0) μ * 2 L *(0) μ * 2 L 0.6 1.5 0.6 0.7 0.8 0.9 * 0.5489 0.3748 0.2828 - -22.1500 -41.1948 -69.5345 * 0.5667 0.3911 0.2961 - -21.9853 -40.7427 -68.6579 * 0.5832 0.4065 0.3089 - -21.8349 -40.3205 -67.8305 * 0.5984 0.4212 0.3212 - -21.6969 -39.9255 -67.0482 2.0 0.6 0.7 0.8 0.9 * 0.5667 0.3911 0.2961 - -21.9853 -40.7427 -68.6579 * 0.5984 0.4212 0.3212 - -21.6969 -39.9255 -67.0482 * 0.6258 0.4484 0.3445 - -21.4529 -39.2068 -65.6050 * 0.6497 0.4732 0.3663 - -21.2436 -38.5697 -64.3036 0.8 1.5 0.6 0.7 0.8 0.9 0.6356 0.5141 0.4277 0.3613 -0.3271 -13.7064 -32.5814 -60.2711 0.6443 0.5234 0.4368 0.3698 -0.2294 -13.4806 -32.1549 -59.5015 0.6525 0.5324 0.4456 0.3780 -0.1367 -13.2648 -31.7456 -58.7621 0.6604 0.5411 0.4542 0.3861 -0.0486 -13.0583 -31.3527 -58.0511 2.0 0.6 0.7 0.8 0.9 0.6443 0.5234 0.4368 0.3698 -0.2294 -13.4806 -32.1549 -59.5015 0.6604 0.5411 0.4542 0.3861 -0.0486 -13.0583 -31.3527 -58.0511 0.6752 0.5574 0.4705 0.4016 0.1153 -12.6709 -30.6119 -56.7079 0.6887 0.5727 0.4859 0.4163 0.2646 -12.3144 -29.9258 -55.4606 178 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion .... Table 5. Pe cen age ela i e loss ** 1 L in he p ecision o he es ima o T** wi h espec o 1 ξ No e: ‘*’ indica es **(0) μ does no exis . Table 6. Pe cen age ela i e loss ** 2 L in he p ecision o he es ima o T** wi h espec o 2 ξ No e: ‘*’ indica es **(0) μ does no exis . W 0.05 0.10 0.15 0.20 ρyx 2 yz **(0) μ ** 1 L **(0) μ ** 1 L **(0) μ ** 1 L **(0) μ ** 1 L 0.6 1.5 0.6 0.7 0.8 0.9 0.8515 0.4376 0.2964 0.2250 3.2070 2.3271 2.0922 2.1709 0.6634 0.3382 0.2305 0.1772 5.8247 4.1649 3.7242 3.8495 0.4538 0.2312 0.1600 0.1257 7.7922 5.5017 4.9027 5.0602 0.2206 0.1163 0.0847 0.0707 9.0181 6.3162 5.6273 5.8180 2.0 0.6 0.7 0.8 0.9 0.6634 0.3382 0.2305 0.1772 5.8247 4.1649 3.7242 3.8495 0.2206 0.1163 0.0847 0.0707 9.0181 6.3162 5.6273 5.8180 * * * * - - - - * * * * - - - - 0.8 1.5 0.6 0.7 0.8 0.9 0.5785 0.4668 0.3883 0.3288 2.3221 2.2289 2.2786 2.4916 0.5297 0.4284 0.3573 0.3038 4.3575 4.1749 4.2545 4.6280 0.4801 0.3891 0.3254 0.2776 6.1286 5.8623 5.9574 6.4507 0.4298 0.3490 0.2925 0.2504 7.6552 7.3121 7.4125 7.9947 2.0 0.6 0.7 0.8 0.9 0.5297 0.4284 0.3573 0.3038 4.3575 4.1749 4.2545 4.6280 0.4298 0.3490 0.2925 0.2504 7.6552 7.3121 7.4125 7.9947 0.3272 0.2666 0.2245 0.1934 10.0446 9.5722 9.6656 10.3593 0.2222 0.1818 0.1538 0.1333 11.6475 11.0819 11.1591 11.9082 W 0.05 0.10 0.15 0.20 yx 2 yz **(0) μ ** 2 L **(0) μ ** 2 L **(0) μ ** 2 L **(0) μ ** 2 L 0.6 1.5 0.6 0.7 0.8 0.9 0.8515 0.4376 0.2964 0.2250 -6.8086 -19.4844 -38.7159 -66.7641 0.6634 0.3382 0.2305 0.1772 -3.9201 -17.2361 -36.4038 -63.9026 0.4538 0.2312 0.1600 0.1257 -1.7490 -15.6008 -34.7341 -61.8388 0.2206 0.1163 0.0847 0.0707 -0.3963 -14.6045 -33.7074 -60.5469 2.0 0.6 0.7 0.8 0.9 0.6634 0.3382 0.2305 0.1772 -3.9201 -17.2361 -36.4038 -63.9026 0.2206 0.1163 0.0847 0.0707 -0.3963 -14.6045 -33.7074 -60.5469 * * * * - - - - * * * * - - - - 0.8 1.5 0.6 0.7 0.8 0.9 0.5785 0.4668 0.3883 0.3288 1.9021 -11.4032 -29.9953 -57.0593 0.5297 0.4284 0.3573 0.3038 3.9462 -9.1858 -27.3668 -53.6183 0.4801 0.3891 0.3254 0.2776 5.7249 -7.2633 -25.1016 -50.6823 0.4298 0.3490 0.2925 0.2504 7.2581 -5.6113 -23.1658 -48.1954 2.0 0.6 0.7 0.8 0.9 0.5297 0.4284 0.3573 0.3038 3.9462 -9.1858 -27.3668 -53.6183 0.4298 0.3490 0.2925 0.2504 7.2581 -5.6113 -23.1658 -48.1954 0.3272 0.2666 0.2245 0.1934 9.6578 -3.0360 -20.1687 -44.3867 0.2222 0.1818 0.1538 0.1333 11.2676 -1.3158 -18.1818 -41.8919 STATISTICS IN TRANSITION new se ies, June 2016 179 9. In e p e a ions o esul s The ollowing conclusions may be d awn om Tables 1-6: (1) F om Tables 1 and 5 i is clea ha (a) Fo he ixed alues o W, ρyx and 2, he alues o (0) **(0) μμ, dec ease wi h he inc easing alues o ρyz. This implies ha he highe he alue o ρyz, he lowe he ac ion o a esh sample equi ed on he cu en occasion. (b) Fo he ixed alues o W, ρyx and ρyz, he alues o (0) **(0) μμ, dec ease and L1, ** 1 L inc ease wi h he inc easing alues o 2. (c) Fo he ixed alues o W, ρyz and 2, no pa e n is obse ed wi h he inc easing alues o ρyx. (d) Fo he ixed alues o 2, ρyz and ρyx, he alues o (0) **(0) μμ, dec ease and L1, ** 1 L inc ease wi h he inc easing alues o W. This beha iou shows ha wi h he highe non- esponse a e one may equi e o d aw he smalle sample on he cu en occasion, which educes he cos o a su ey. (2) F om Tables 2 and 6 i may be seen ha (a) Fo he ixed alues o W, ρyx and 2, he alues o (0) **(0) μμ, and L2, ** 2 L dec ease wi h he inc easing alues o ρyz.. This implies ha i one uses he in o ma ion on highly co ela ed auxilia y a iable, he e is a signi ican gain in he p ecision o es ima es. (b) Fo he ixed alues o W, ρyx and ρyz, he alues o (0) **(0) μμ, dec ease and L2, ** 2 L inc ease wi h he inc easing alues o 2. (c) Fo he ixed alues o W, ρyz and 2, he alues o L2, ** 2 L inc ease wi h he inc easing alues o ρyx. (d) Fo he ixed alues o 2, ρyz and ρyx, he alues o (0) **(0) μμ, dec ease and L2, ** 2 L inc ease wi h he inc easing alues o W. This pa e n shows ha he highe he non- esponse a e, he g ea e he loss. This beha iou is p ac ically jus i ied. (3) F om Table 3 i is clea ha (a) Fo he ixed alues o W, ρyx and 2, he alues o *(0) μ dec ease and * 1 L inc ease wi h he inc easing alues o ρyz. This beha iou indica es ha i he in o ma ion on highly co ela ed auxilia y a iable is a ailable, i plays an impo an ole in imp o ing he p ecision o es ima es. (b) Fo he ixed alues o W, ρyx and ρyz, he alues o *(0) μ and * 1 L inc ease wi h he inc easing alues o 2. 180 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion .... (c) Fo he ixed alues o W, ρyz and 2, no pa e n is isible wi h he inc easing alues o ρyx. (d) Fo he ixed alues o 2, ρyz and ρyx, he alues o *(0) μ and * 1 L inc ease wi h he inc easing alues o W. (4) F om Table 4 i may be seen ha (a) Fo he ixed alues o W, ρyx and 2, he alues o *(0) μ and * 2 L dec ease wi h he inc easing alues o ρyz. This implies ha nega i e loss is obse ed due o he p esence o high co ela ion be ween he auxilia y a iables. This beha iou is highly desi able. (b) Fo he ixed alues o W, ρyx and ρyz, he alues o *(0) μ and * 2 L inc ease wi h he inc easing alues o 2. This indica es ha i a smalle size o sub-sample is chosen, he loss in p ecision inc eases, as i was expec ed. (c) Fo he ixed alues o W, ρyz and 2 no pa e n is seen wi h he inc easing alues o ρyx. (d) Fo he ixed alues o 2, ρyz and ρyx, he alues o *(0) μ and * 2 L inc ease wi h he inc easing alues o W. 10. Conclusions and ecommenda ions I may be seen om he abo e ables ha o all cases he pe cen age ela i e loss in p ecisions is obse ed whe e e he op imum alue o μ exis s, when non- esponse occu s on bo h occasions. F om Tables 1, 3 and 5, i is seen ha he loss is p esen due o he p esence o non- esponse on each occasion, bu he nega i e impac o non- esponse is e y low, which jus i ies he use o Hansen and Hu wi z (1946) echnique in he p oposed es ima ion p ocedu es. F om Tables 2, 4 and 6, when he p oposed es ima o s a e compa ed wi h he na u al successi e sampling es ima o , subs an ial p o i is isible, which jus i ies he in elligible use o auxilia y in o ma ion in he o m o exponen ial me hods o es ima ion. Finally, looking a good beha iou s o he p oposed es ima o s one may ecommend hem o su ey s a is icians and p ac i ione s o hei p ac ical applica ions. 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