COMSATS Institute of Information and Technology, Lahore Pakistan 2

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1 Pak. J. Stati. 015 Vol. 31(1), GENERAIZED EXPONENTIA-TPE RATIO-CUM-RATIO AND PRODUCT-CUM-PRODUCT ESTIMATORS FOR POPUATION MEAN IN THE PRESENCE OF NON-RESPONSE UNDER STRATIFIED TWO-PHASE RANDOM SAMPING Aamir Sanaullah 1, Muhammad Noor-ul-Amin 1 and Muhammad Hanif 1 COMSATS Initute of Information and Technology, ahore Pakian chaamirsanaullah@yahoo.com; nooramin.ats@gmail.com National College of Business Adminiration and Economics, ahore, Pakian. drmianhanif@gmail.com Corresponding author ABSTRACT In this paper, generalized exponential-type eimators have been proposed for eimating the finite population mean of udy variable using information on two auxiliary variables in the presence of non-response under ratified two-phase random sampling. The expressions for the bias and mean square error (MSE) of proposed eimators have been derived in two different situations of non-response. Theoretical comparisons of proposed eimators have been made with modified forms of Hansen and Hurwitz (1946), ratio and product eimators to the ratified two-phase sampling method. An empirical udy has also been carried out to demonrate the performances of proposed eimators. KEWORDS Auxiliary variables; Bias; Mean Square error; Non-response; Stratified Two-phase Random Sampling; Exponential-type eimator. AMS (1991) subject classification. 6D05 1. INTRODUCTION Eimation of the population mean in sample surveys when some observations are missing due to non-response has been considered by Hansen and Hurwitz (1946). In the presence of non-response, the problem of eimating population mean of the udy variable (y) has been discussed by Cochran (1977), Khare and Srivaava (1997) and Singh and Kumar (008). Singh and Kumar (009) proposed a general family of eimators under two-phase sampling in the presence of non-response. Singh et al. (009) have proposed a family of combined-ratio-type eimators for eimating the population mean under non-response by adapting the eimator of Khoshnevisan et al. (007). Khare and Sinha (009) proposed some eimators using multi-auxiliary characteriics with known population means in the presence of non-response. Singh et al. (010) proposed some exponential-type ratio and product type eimators and their generalized version for eimating the population mean of the udy variable (y) in the presence of non-response. Singh et al. (010) suggeed some exponential-type ratio type eimators using single auxiliary variable under two-phase sampling in the presence of non-response. 015 Pakian Journal of Statiics 71

2 7 Generalized exponential-type ratio-cum-ratio and product-cum Kadilar and Cingi (003) extended Upadhyaya and Singh (1999) eimator of simple random sampling in ratified random sampling. Singh et al. (008, 009) proposed some exponential eimators in simple random sampling to eimate the population mean of the udy variable y, following Bhal and Tuteja (1991). Singh et al. (008) have considered exponential-ratio-type eimator in ratified random sampling. Koyuncu and Kadilar (010) have suggeed a family of eimators in ratified random sampling following Diana (1993) and Kadilar and Cingi (003). Upadhyaya et al. (011) suggeed some improved ratio and product exponential type eimators. Malik and Singh (01) proposed some modified ratio type eimators using geometric mean and harmonic mean for ratified random sampling. Motivated by Singh et al. (009), Singh and Kumar (01) proposed some exponential eimators in ratified sampling. et Consider a finite population of size N N h be the size of th which is ratified into homogenous rata. h ratum ( h 1,,..., ) such that h1 Nh N and hi, hi the observations of the udy variable ( y ) and the auxiliary variable x, on the th of h ratum, respectively. et, y h and corresponding to the population means x h be the sample means of th y x be h th i unit ratum h and approximations to the bias and mean square error (MSE) for the proposed eimators under ratified two-phase sampling, let us define notations and expectations for situation-i and situation-ii separately as, X h respectively. In order to obtain Situation-I: When non-response is observed on all the variables taken at secondphase. Situation-II: When the udy variable is observed with non-response at second-phase whereas the two auxiliary variable(s) are observed with complete response. i) Notations for Situation-I yh h xh X h zh Zh oh 1 h h h X h Zh e, e, e, Ph h eoh Ph X he1 h Ph Zheh h1 h1 h1 o, 1, e e e X Z n n y n y x n z n z y x z h hi h(1) n h() r i1 h(1) n h() r h h and h nh nh nh h(1) h h(1) h kh 1 h, h h Wh() h h h nh nh nh Nh nh N N P W S S y N i h() y () N () h h h i h h h() hy and hy N Nh i1 Nh i1 Nh() () 1 1 (1.1)

3 Sanaullah, Amin and Hanif 73 ii) Notations for Situation-II n h Ph Zheh zhi zh Z h h1 i1 eh, e and zh (1.) Zh Z nh where e i in (1.1) and (1.) for i x, y,and z is the sampling error. Further we assume that and E( eo ) E( e1 ) E( e) 0, E( e1 ) E( e) 0 respectively for Situation-I and Situation-II. (1.3) In ratified random sampling without replacement, we may obtain the following expectations respectively for Situation-I and Situation-II of non-response as, iii) Expectations for Situation-I r s t h h h h h h E x X y z Z, r Vr,, s t P h r s t h1 X Z 1 1 E( eo ) Ph hs yh hs yh V00 E( e1 ) Ph h Sxh V00 h1 X h1 1 1 E( e ) P h hszh hszh V00 E( eo. e ) Ph hs yzh hs yzh V011 Z h1 Z h1 1 1 E( eo. e1 ) Ph h S yxh V E( e1. e ) Ph h Sxzh V101 X h1 ZX h1 00 V00 V00 V V iv) Expectations for Situation-II 1 l 1 l E( eo ) Ph hsyh hsyh V00 E( e1 ) Ph h Sxh V00 h1 X h1 1 1 E( e ) P S V E( e. e ) P S S V Z 1 1 E( e. e ) P S V E( e. e ) P S l l h h zh 00 o h h yzh h yzh 011 h1 Z h1 l l o 1 h h yxh 1 h h xzh V101 X h1 ZX h1 (1.4) 00 V00 V00 V V (1.5)

4 74 Generalized exponential-type ratio-cum-ratio and product-cum. NON-RESPONSE IN STRATIFIED TWO-PHASE SAMPING In ratified two-phase sampling, a fir phase sample of size n h from the h ratum using simple random sampling without replacement (SRSWOR) is selected such that h1nh n and observe auxiliary characteriic(s) for these units. A second phase sample of size n h ( nh n h) is selected SRSWOR such that h1nh n and collect information on the udy variable say y. From the available second-phase sample n h, only nh(1) units respond to the survey and nh() do not respond. From n h() non-respondents, nh() a sub-sample of r h rh, kh 1 k units is selected and information are obtained from h these r h units. A modified form of Hansen and Hurwitz (1946) eimator to the ratified two-phase sampling is given as y h1 Ph yh (.1) The eimator (.1) is unbiased with variance, h h hy h h hy 00 h1 h1 Var( y ) P S P S V (.) Following Khare and Srivaava (1993) and Tabasum Khan (004), ratio and product eimators are modified to the ratified two-phase sampling in two different situations of non-response separately as, i) Situation-I x Rd y x and The mean square errors of are, Pd y Rd Pd x x th (.3) Rd and Pd in (.3), up to the fir order approximation MSE V (.4) MSE V (.5) ii) Situation-II Rd x y and x Pd y x x (.6) The mean square errors of Rd are, and Pd in (.6), up to the fir order approximation

5 Sanaullah, Amin and Hanif 75 Rd Pd MSE V (.7) MSE V (.8) In the following section, we propose some generalized eimators of population mean for ratified two-phase sampling with non-response. 3. PROPOSED GENERAIZED EXPONENTIA-TPE ESTIMATORS In this section, motivated by Singh and Kumar (01), and Upadhyaya et al. (011), some generalized exponential-type ratio-cum-ratio and product-cum-product eimators have been suggeed for two different situations of non-response using two auxiliary variables with known means. 3.1 Fir Proposed Generalized Eimator for Situation-I We propose a generalized exponential-type ratio-cum-ratio eimator of population mean for situation-i under ratified two-phase sampling in the presence of non-response as, where a l l l Ph X h xh Ph Zh zh G h1 h1 Ph yh exp exp l l h1 Ph X h a 1 xh Ph Zh ( b 1) zh h1 h1 (3.1) and b are some suitably chosen scalars whose values are to be eimated so that MSE of G is minimized. It is remarked that for various values of a and b in (3.1), we get various exponentialtype ratio eimators as deduced family of below as, G. From this family some are given in i) For a 1 and b 1, 1 is deduced as the family of G as, Ph X h xh Ph Zh zh 1 h1 h1 Ph yh exp exp h1 Ph X h Ph Z h h1 h1 ii) For a and b 1, is deduced as the family of G as, Ph X h xh Ph Zh zh h1 h1 Ph yh exp exp h1 Ph X h x Ph Zh h1 h1 (3.) (3.3)

6 76 Generalized exponential-type ratio-cum-ratio and product-cum iii) For a and b, A is deduced as Singh and Kumar (01) type eimator as the family of G as, Ph X h xh Ph Zh zh A h1 h1 Ph yh exp exp h1 Ph X h xh Ph Zh zh h1 h1 Using (1.1) the eimator in (3.1) may be expressed as, G (1 e0 ) exp e 1G ( a 1) e1 G exp e G ( b 1) e G (3.4) (3.5) Expanding the right hand side of (3.5) up to the second order of approximation, we have e e e a e b e e e e e e e G 0 1 G G ( 1) 1 G ( 1) G 0 1 G 0 G 1 G (3.6) Using (3.6) the bias and the MSE equations of G are obtained as, Bias a1 b a b a b ab G V00 V 00 V V011 V101 G MSE V 1 V 1 V 1 V 1 V 1 V a b a b ab (3.7) (3.8) In order to obtain minimum MSE of G, we differentiate MSE ( G ) in (3.8) with respect to a and b respectively. The optimum value of a and b are obtained as a V V V V V V and b V V V V V V V V (3.9) Subitution for the optimal values of a and b in (3.9) yields the minimum value of MSE( G ) as G min. MSE V V00V V00 V011 V 101 V011V 00 V00 V00 V101 The bias and MSE expressions for deduced family the values of a and b in (3.7) and (3.8) respectively as (3.10) G may be obtained by putting

7 Sanaullah, Amin and Hanif 77 i) For a 1 and b 1, the bias and MSE of 1 Bias is obtained as 1 V V 011 V (3.11) MSE V V V V V V (3.1) ii) For a and b 1, the bias and MSE of Bias V00 V V011 V101 is obtained as 1 MSE V V V 1 V V 1 V iii) For a and b, the bias and MSE of Bias A 101 V00 V00 V V V 4 A 1 1 MSE V V V V V V (3.13) (3.14) (3.15) (3.16) 3. Second Proposed Generalized Eimator for Situation-I We propose another generalized exponential-type product-cum-product eimator of population mean for situation-i under ratified two-phase sampling as, Ph xh X h Ph zh Zh G h1 h1 Ph yh exp exp h1 Ph c 1 xh X h Ph ( d 1) zh Zh h1 h1 (3.17) where c and d are some suitably chosen scalars whose values are to be eimated so that MSE of G is minimized. It is remarked that for various values of c exponential-type ratio eimators as deduced family of given in below as, and d in (3.17), we get various G. From this family some are

8 78 Generalized exponential-type ratio-cum-ratio and product-cum i) For c 1 and d 1, 1 is deduced as the family of G as Ph xh X h Ph zh Zh 1 h1 h1 Ph yh exp exp h1 Ph X h Ph Z h h1 h1 ii) For c and d 1, is deduced as the family of G as Ph xh X h Ph zh Zh h1 h1 Ph yh exp exp h1 Ph X h x Ph Zh h1 h1 (3.18) (3.19) 3 iii) For c and d, is deduced as the family of G as, Ph xh X h Ph zh Zh A h1 h1 Ph yh exp exp h1 Ph xh X h Ph zh Zh h1 h1 Using (1.1) the eimator (3.17) may be expressed as G (1 e0 ) exp e 1G ( c 1) e1 G exp e G ( d 1) e G (3.0) (3.1) Expanding the right hand side of (3.1) up to the second order of approximation, we have e e e c e d e e e e e e e G 0 1 G G ( 1) 1 G ( 1) G 0 1 G 0 G 1 G (3.) Using (3.) the bias and MSE equations of G are obtained as, Bias c1 d c d c d cd G V00 V 00 V V011 V101 G MSE V 1 V 1 V 1 V 1 V 1 V c d c d cd (3.3) (3.4) In order to obtain minimum MSE of G, we differentiate MSE ( G ) in (3.4) with respect to c and d respectively. The optimum value of c and d are obtained as

9 Sanaullah, Amin and Hanif z Rx V V V R V V V c and d V V V V V V V V (3.5) Subitution for the optimal values of c and d in (3.4) yields the minimum value of MSE( G ) as G min. MSE V V00V V00 V011 V 101 V011V 00 V00 V00 V101 (3.6) G The bias and MSE expressions for deduced family may be obtained by putting the values of c and d in (3.3) and (3.4) respectively as i) For c 1and d 1, the bias and MSE of Bias 1 is obtained as, 1 V V 011 V (3.7) MSE V V V V V V (3.8) ii) For c and d 1, the bias and MSE of Bias V00 V V011 V is obtained as, 1 MSE V V V 1 V V 1 V iii) For c and d A, the bias and MSE of is obtained as, Bias A 101 V00 V00 V V V 4 A 1 1 MSE V V V V V V (3.9) (3.30) (3.31) (3.3) 3.3 Fir Proposed Generalized Eimator for Situation-II We propose a generalized exponential-type eimator of population mean for situation-ii under ratified two-phase sampling as,

10 80 Generalized exponential-type ratio-cum-ratio and product-cum l l l Ph X h xh Ph Zh zh G h1 h1 Ph yh exp exp l l h1 Ph X h a 1 x h Ph Zh ( b 1) z h h1 h1 (3.33) where a and b are some suitably chosen scalars whose values are to be eimated so that MSE of G is minimized. It is remarked that for various values of a and b in (3.33), we get various G exponential-type ratio eimators as deduced family of. From this family some are given in below as, i) For a 1 and b 1, 1 is deduced as the family of G as Ph X h xh Ph Zh zh 1 h1 h1 Ph yh exp exp h1 Ph X h Ph Z h h1 h1 (3.34) ii) For a and b 1, is deduced as the family of G as Ph X h xh Ph Zh zh h1 h1 Ph yh exp exp h1 Ph X h x Ph Zh h1 h1 (3.35) iii) For a and b, A is deduced as Singh and Kumar (01) as the family of G for situation-ii. Ph X h xh Ph Zh zh A h1 h1 Ph yh exp exp h1 Ph X h x h Ph Zh z h h1 h1 (3.36) The proposed eimator in (3.33) follows naturally in exactly the same fashion as that for Situation-I in Section 3.1. In addition, the relation between a and b is the same as that for Situation-I in Section 3.1. Finally, the same is true for the MSE and the bias. It is therefore directly from Section 3.1 we may have the bias and the MSE equations of G are obtained as, Bias a 1 b a b a b a b G V00 V 00 V V011 V101 (3.37)

11 Sanaullah, Amin and Hanif 81 G 1 1 MSE V V V 1 V 1 V 1 V a b a b a b G MSE is minimized for the optimal values of a a and b given as V V V V V V and b V V V V V V V V (3.38) (3.39) Subitution for the optimal values of a and b in (3.38) yields the minimum value of G MSE as min. MSE G V00V V00 V011 V 101 V011V V00 V00V 00 V 101 The bias and MSE expressions for deduced family the values of a and b in (3.37) and (3.38) respectively. (3.40) G may be obtained by putting 3.4 Second Proposed Generalized Eimator for Situation-II We propose another generalized exponential-type product-cum-product eimator of population mean for situation-ii as Ph xh X h Ph zh Zh G h1 h1 Ph yh exp exp h1 Ph c 1 xh X h Ph ( d 1) zh Z h h1 h1 (3.41) It is remarked that for various values of c and d in (3.41), we get various G exponential-type ratio eimators as deduced family of. From this family some are given in below as, i) For c 1 and d 1, 1 is deduced as the family of G as Ph xh X h Ph zh Zh 1 h1 h1 Ph yh exp exp h1 Ph X h Ph Z h h1 h1 (3.4)

12 8 Generalized exponential-type ratio-cum-ratio and product-cum ii) For c and d 1, is deduced as the family of G as Ph xh X h Ph zh Zh h1 h1 Ph yh exp exp h1 Ph X h x Ph Zh h1 h1 (3.43) iii) For c and d, A is deduced as Singh and Kumar (01) as the family of G for situation-ii. Ph xh X h Ph zh Zh A h1 h1 Ph yh exp exp h1 Ph xh X h Ph zh Z h h1 h1 (3.44) The proposed eimator in (3.41) follows naturally in exactly the same fashion as that for Situation-I in Section 3.. In addition, the relation between c and d is the same as that for Situation-I in Section 3.. Finally, the same is true for the MSE and the bias. It is therefore directly from Section 3., we may have the bias and the MSE equations of G are obtained as, Bias c 1 d c d c d c d G V00 V 00 V V011 V101 G (3.45) MSE V00 V 00 V 00 V V011 V101 c d c d c d (3.46) In order to obtain minimum MSE of G G, we differentiate MSE in (3.46) with respect to c and d respectively. The optimum value of c and d are obtained as c z Rx V V V R V V V and and d V V V V V V V V (3.47) Subitution for the optimal values of c and d in (3.45) yields the minimum value of G MSE as G min. MSE( ) V V00V V00 V011 V 101 V011V 00 V00 V00 V101 (3.48)

13 Sanaullah, Amin and Hanif 83 The bias and MSE expressions for deduced family the values of c and d in (3.45) and (3.46) respectively. 4. EFFICIENC COMPARISONS G may be obtained by putting Now we compare the proposed generalized eimators with usual Hansen and Hurwitz s (1946) unbiased eimator y and some other considered eimators under two different cases as i) Ratio-cum-ratio Eimators under Situation-I 1 MSE( ) MSE( y ) V00 V00 V011 V101 if 1 V and V00 V00 00 if V V011 V101 MSE( ) ( ) A MSE( ) MSE( y ) 1 MSE Rd V00 V00 V011 V101 if 1 V A MSE( ) MSE( ) and V00 V V V011 V101 Rd if 1 MSE( ) ( ) MSE y V00 4V00 V011 V101 if 1 4 V and V00 4V V V011 V101 MSE( ) ( ) MSE Rd if 1 (4.1) (4.) (4.3)

14 84 Generalized exponential-type ratio-cum-ratio and product-cum G MSE( ) MSE( y ) V101 V011V if 1 4 V V V V and 00 V101 -V00 V00 V V V V V V V G MSE( ) MSE( ) Rd if ii) Product-cum-product Eimators under Situation-I 1 MSE( ) MSE( y ) 00 V00 V if 1 V V011 V101 and 00 V00 V00 V V011 V101 MSE( ) ( ) 1 MSE Pd if 1 A MSE( ) MSE( y ) 00 V00 V if 1 V V011 V101 A MSE( ) MSE( ) and 4 00 V00 V00 V V011 V101 Pd if 1 MSE( ) ( ) MSE y 00 V00 V 4 if 1 4 V V011 V101 and 4 00 V00 4V00 4 V V011 V101 MSE( ) ( ) MSE Pd if 1 (4.4) (4.5) (4.6) (4.7)

15 Sanaullah, Amin and Hanif 85 G MSE( ) MSE( y ) V101 V011V if 1 V V V V and 00 V101 -V00 V00 V V V V V V V G MSE( ) MSE( ) Pd if iii) Ratio-cum-ratio Eimators under Situation-II MSE( ) ( ) 1 MSE y V00 V00 if 1 V V V and V00 V00 00 if V V011 V101 MSE( ) ( ) A MSE( ) MSE( y ) 1 MSE Rd V00 V00 if 1 V V V A MSE( ) MSE( ) and V00 V V V011 V101 Rd if 1 MSE( ) ( ) MSE y V00 4V00 if 1 4 V V V and V00 4V V V011 V101 MSE( ) ( ) MSE Rd if 1 (4.8) (4.9) (4.10) (4.11)

16 86 Generalized exponential-type ratio-cum-ratio and product-cum G MSE( ) MSE( y ) V V V if 1 V V V V and 00 V101 -V00 V00 V V V V V V V G MSE( ) MSE( ) Rd if iv) Product-cum-product Eimators under Situation-II 1 MSE( ) MSE( y ) V00 V00 if 1 V V V and 00 V00 V00 V V011 V101 MSE( ) ( ) 1 MSE Pd if 1 A MSE( ) MSE( y ) V00 V00 if 1 V V V A MSE( ) MSE( ) and 4 00 V00 V00 V V011 V101 Pd if 1 MSE( ) ( ) MSE y V00 4V00 if 1 4 V V V and 4 00 V00 4V00 4 V V011 V101 MSE( ) ( ) MSE Pd if 1 (4.1) (4.13) (4.14) (4.15)

17 Sanaullah, Amin and Hanif 87 G MSE( ) MSE( y ) V V V if 1 V V V V and 00 V101 -V00 V00 V V V V V V V G MSE( ) MSE( ) Pd if (4.16) 5. EMPIRICA STUD In order to examine the performance of proposed eimators under ratified twophase sampling, we have taken two different ratified populations: Population-I: (source: Koyuncu and Kadilar (009)) We consider No. of teachers as udy variable (), No. of udents as auxiliary variable (X), and No. of classes in primary and secondary schools as another auxiliary variable (Z) for 93 diricts at six 6 regions (1: Marmara, : Agean, 3: Mediterranean, 4: Central Anatolia, 5: Black Sea, and 6: Ea and Southea Anatolia) in Turkey in 007. Population-II: (source: detailed livelihood assessment of flood affected diricts of Pakian September 011, Food Security Cluer, Pakian) We consider food expenditure as udy variable (), household earn as auxiliary variable (X), and total expenditure in May (011) as another auxiliary variable (Z) for (6940) male and (1678) female households in flood affected diricts of Pakian. The Neyman allocation has been used for allocating the samples to different rata. The comparison of proposed generalized exponential-type ratio-cum-ratio and exponential-type product-product eimators have been made with respect to Hansen and Hurwitz s (1946), ratified two-phase ratio and ratified two-phase product eimators. The information for two populations is given in appendix-a Table CONCUSION The MSE values of the eimators are computed in Table 3-4 using (.), (.4), (.5), (.7), (.8), (3.10), (3.1), (3.14), (3.16), (3.6), (3.8), (3.30), (3.3), (3.38), (3.40), (3.46), and (3.48). Table 1- indicates that the performance of exponential-type ratiocum-ratio eimators in both situations is better than modified ratified two-phase due to Hansen and Hurwitz s (1946) eimator y. The generalized exponential-type ratio-cumratio eimators are more efficient as compared to ratio eimators Rd, Rd modified to the ratified two-phase for both situations of non-response. Furthermore, it is observed that proposed eimators are more efficient than ratio eimators for both situations of non-response Rd, Rd.

18 88 Generalized exponential-type ratio-cum-ratio and product-cum The PREs for the proposed family exponential-type eimators increases if inverse sampling rate k h increases from.0 to 3.5 at Wh 0% and Wh 30% but this is not A true for. For Wh 10%, PREs of the eimators decreases in situation-i of nonresponse. In situation-ii of non-response, the PREs for Singh and Kumar (01) type eimators including ratified two-phase ratio eimator decreases if the inverse sampling rate kh increases from.0 to 3.5 at each of higher non-response. It is found from Table 1- that generalized exponential-type product-cum-product eimators are more efficient than modified ratified two-phase Hansen and Hurwitz s (1946) eimator Pd, Pd, Pd in both y and ratified two-phase ratio eimators situations of non-response. When we further observe we can infer on the basis of PREs that performance of generalized exponential-type eimators in section 3 is better in both situations of non-response than the performance of the Singh and Kumar (01) type eimators. Furthermore we can see that generalized exponential-type eimators are more efficient in situation-i than the generalized exponential-type eimators in situation-ii. Finally we conclude that the class of generalized exponential-type eimators can be proposed for their practical application for both of the situations of non-response. 7. REFERENCES 1. Bahl, S. and Tuteja, R.K. (1991). Ratio and product type exponential eimators. Information and Optimization Sciences, 1(1), Chaudhary, M.K., Singh, R., Shukla, R.K., Kumar, M. and Smarandache, F. (009). A family of eimators for eimating population mean in ratified sampling under non-response. Pak. J. Stat. Oper. Res, 5(1), Cochran, W.G. (1977). Sampling Techniques. 3rd edition, John Wiley: New ork. 4. Diana, G. (1993). A class of eimators of the population mean in ratified random sampling. Statiica, 53(1), Hansen, M.H. and Hurwitz, W.N. (1946). The problem of non-response in sample surveys. J. Amer. Stati. Assoc., 41, Kadilar, C. and Cingi, H. (003). Ratio eimator in ratified sampling. Biometrical Journal, 45, Khare, B.B. and Srivaava, S. (1997). Transformed ratio type eimators for the population mean in the presence of non-response. Comm. Stat.-Theory Methods, 6(7), Khare, B.B. and Sinha, R.R. (009). On class of eimators for population mean using multi-auxiliary characters in the presence of non-response. Statiics in Transition- New Series, 10(1), Khoshnevisan, M., Singh, R., Chauhan, P., Sawan, N. and Smarandache, F. (007). A general family of eimators for eimating population mean using known value of some population parameter(s). Far Ea Journal of Theoretical Statiics,, Koyuncu, N. and Kadilar, C. (009): Ratio and product eimators in ratified random sampling. J. Stati. Plann. and Infer., 139(8), Koyuncu, N. and Kadilar, C. (010). On the family of eimators of population mean in ratified random sampling. Commun. in Stati.-Theo. and Meth., 6(),

19 Sanaullah, Amin and Hanif Malik, S. and Singh, R. (01). Some improved multivariate ratio type eimators using geometric and harmonic means in ratified random sampling. ISRN Prob. and Stat. Article ID , doi:10.540/01/ Singh, H.P., Kumar, S. and Kozak, M. (010). Improved eimation of finite population mean using sub-sampling to deal with non-response in two-phase sampling scheme. Commun. in Stati.-Theo. and Meth., 39(5), Singh, H.P. and Kumar, S. (008). A general family of eimators of finite population ratio, product and mean using two phase sampling scheme in presence of non response. J. Stati. Theo. & Practice, (4), Singh, H.P. and Kumar, S. (009). A general procedure of eimating the population mean in the presence of non-response under double sampling using auxiliary information. SORT, 33(1), Singh, R. and Kumar, M. (01). Improved eimators of population mean using two auxiliary variables in ratified random sampling. Pak. J. Stat. Oper. Res., 8(1), Singh, R., Chauhan, P., Sawan, N. and Smarandache, F. (009). Improvement in eimating the population mean using exponential eimator in simple random sampling. Bull. of Stat. and Econ., 3, Singh, R., Kumar, M., Singh, R.D. and Chaudhary, M.K. (008). Exponential ratio type eimators in ratified random sampling, Presented in International Symposium on Optimisation and Statiics at A.M.U., Aligarh, India, during 9-31 Dec Tabasum, R. and Khan, I.A. (004). Double sampling for ratio eimation with nonresponse. J. Ind. Soc. Agril. Stati., 53(3), Upadhyaya,.N. and Singh, H.P. (1999). Use of transformed auxiliary variable in eimating the finite population mean. Biometrical Journal, 41, Upadhyaya,.N., Singh, H.P., Chatterjee, S. and aday, R. (011). Improved ratio and product exponential type eimators. Jour. Stat. Ther. and Practice, 5(),

20 Population No 90 Generalized exponential-type ratio-cum-ratio and product-cum Table 1: Percent Relative Efficiencies (PREs) of Eimators with respect to for Different Values of kh each at Different Rate of Non-Response under Situation-I using Two Different Populations AENDIX y W h k h y Rd A 1 Pd A 1 G G 10% 0% 30% () shows population is not applicable.

21 Population No Sanaullah, Amin and Hanif 91 Table : Percent relative efficiencies (PREs) of eimators with respect to for different values of kh each at different rate of non-response under situation-ii using two different populations y W h k h y Rd A 1 Pd A 1 G G 10% 0% 30% () shows population is not applicable.

22 Population No 9 Generalized exponential-type ratio-cum-ratio and product-cum Table 3: MSEs of y, Rd, Pd and Proposed Eimators for Different Values of kh each at Different Rate of Non-Response under Situation-I Using Two Different Populations W h k h y Rd A 1 Pd A 1 G G 10% 0% 30% () shows population is not applicable.

23 Population No Sanaullah, Amin and Hanif 93 Table 4: MSEs of y, Rd, Pd and Proposed Eimators for Different Values of kh each at Different Rate of Non-Response under Situation-II using Two Different Populations W h k h y Rd A 1 Pd A 1 G G 10% 0% 30% () shows population is not applicable.

24 Wh=30% Non-response Wh=0% Non-response Wh=10% Non-response Stratified Mean, S.Ds and Correlation Coefficients 94 Generalized exponential-type ratio-cum-ratio and product-cum Table 5: Data Sets Population-I Population-II Stratum (h) N h n h n h S yh S xh S zh h X h Z h xyh xzh yzh S yh S xh S zh xy xz yz S yh S xh S zh xy xz yz S yh S xh S zh xy xz yz