International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 Variance Estimation in Stratified Random Sampling in te Presence of Two Auxiliary Random Variables Esubalew Belay Sidelel, George Otieno Orwa, Romanus Odiambo Otieno 3 Department of Matematics and Statistics, Basic Science, Tecnology and Innovation, Pan African University, JKUAT, Kenya Statistics and Actuarial Science Department, Jomo Kenyatta University of Agriculture and Tecnology (JKUAT), Kenya 3 Statistics and Actuarial Science Department, Jomo Kenyatta University of Agriculture and Tecnology (JKUAT), Kenya Abstract: Te objective of tis paper is to develop an improved population variance estimators in te presence of two auxiliary variables in stratified random sampling adapting te family of estimators proposed by Koyuncu and Kadilar (009) for te estimation of population mean in stratified random sampling using prior information of te two auxiliary variables. In tis paper, we proposed ratio-product type estimators and derived teir mean square errors using first order approximation of Taylor series metod. Efficiency comparisons of proposed estimators wit respect to teir mean square errors ave been discussed and acieved improvement under certain conditions. Results are also supported by numerical analysis. Based on results, te proposed ratio- type variance estimators may be preferred over traditional ratio-type and sample estimator of population variance for te use in practical applications. Keywords: Variance estimator; Ratio-product type estimators, Mean square error, Auxiliary information; Efficiency; Stratified random sampling. Introduction In sample surveys, it is well known tat to use information of auxiliary variable(s) to estimate unknown population parameter(s) in various sampling designs. In sampling literature, many Autors ave used information of auxiliary variables suc as population mean, variance, kurtosis, skewness, etc to estimating population mean and variance of te study variable. Many autors wo done important work in tis area, were Das and Tripati (978), Srivastavaetal and Jajji (980,983,995), Isakietal (983,000), Sing and Kataria (990), Prasad and Sing (990,99), Amedetal.(000,003), Gupta and Sabbir (006). Kadilar and Cingi (006) studied population variance of interest variable using population mean, variance, kurtosis and coefficient of variation of auxiliary variable in simple and stratified random sampling. Recently Olufadi and Kadilar (04) estimated te population variance of interest variate in simple and two-pase sampling by using te variance of auxiliary variables and got interesting results. Tis paper mainly focuses on population variance estimators using prior knowledge of two auxiliary variables in stratified sampling design. Consider a finite population P =,,,, of N units. Let te study and two auxiliary variables are denoted by Y, X and Z associate wit eac (j=,,n) of te population respectively. Let te population is stratified in to K strata wit stratum containing units, were =,,3,,K suc tat =N and from te stratum, a sample is drawn by simple random sampling witout replacement suc tat =n. Let (,, ) denote te observed values of Y, X, and Z on te unit of te stratum were i=,,,. Te population variance of te study variable (y) and te auxiliary variables are defined as follows. Volume 3 Issue 9, September 04 ( ), = ( ) = ( ) + ( ) Were te population is mean of te variate of interest in stratum, and is te value of te observation of interest variate in stratum. For large sample size, assuming tat and, ten, + ( ) = - sample mean of stratum. Paper ID: SEP4658 453 = - is te sample estimator of population mean of te study variable. = -population mean of stratum. = ( ) - population variance of stratum. = ( ) -sample estimator of population variance in te stratum. Similar expression are defined for te auxiliary variables x and z.. Adapted estimators Koyuncu and Cem Kadilar (009), defined te classical ratio estimator to estimate te population mean of te study variable Y in te stratified random sampling wen tere are two auxiliary variables as follows: = -------------------------- () Were and are te population mean of te two auxiliary variables and, and are sample estimate of te population mean in stratified random sampling sceme. Te regression estimator of te population mean also defined as: = + ( ) + ( ) ------ () Were = and =. Adapting te estimator given in () and () to te estimator for te population variance of te study variable y and assuming te population variance of te two auxiliary variables in eac stratum is known, we
International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 develop te following ratio -product type and regression estimators: =, ---------------------------- (3),, =,+ ( -,)+ ( -,) ------(4) Were,= + ( ),,= + ( ), and,= + ( ), are te sample estimator of population variance of eac variables in stratified sampling sceme wen neglecting population correction factor of eac stratum. Te mean square error of te variance estimator, given in (3) and (4), is obtained as follows: + --------------- (5) ( ) H H H H 3 + -------------------- (6) see Appendix (A. 3) and (B. ) 3. Te Proposed Estimators In tis section some variance estimators are proposed using te variance of two auxiliary variables, population kurtosis, coefficient of variation and teir combination. Motivated by Cingi and Kadilar (005a, 006b) and Koyuncu and Kadilar (009), te following population variance estimators are proposed in te stratified random sampling: =, =, () (), (), () () (), (), () ----------------- (7) ------------ (8) Te MSE of te estimators, given in (7) and (8) is found using te first degree approximation of Taylor series metod as follows: () () ()( () ) H + H - (9) 4 () () ()( () ) H + H -- (0) 5 (See Appendix C and D) were and - are population coefficient of variation of te auxiliary variables (X) and (Z) respectively. () and () are te population kurtosis of te auxiliary variables (X) and (Z) respectively. Te detail derivations of all te mean square error of te estimators considered in tis paper was presented in appendix at te end of te paper. 4. Efficiency Comparison of te Estimators In tis section, we compare te performance of te proposed estimators wit oter estimators considered ere and some efficiency comparison condition is carry out under wic te proposed estimators are more efficient tan te usual sample estimator of population variance and te adapted variance estimators considered in tis paper. Tese conditions are given as follows: ( )-, <0 if < --- () -, <0 if <0 -------------- () -,<0 if < () () () () ()( () ) ---- (3) -,<0 < Table : Data statistics () () () () ()( () ) (4) Were for i=,3,4,5- is te term of eac mean square error wit out te common multiplier of all terms and. Te oter metod wic is used to compare te performance of te proposed estimators over is Percent Relative Efficient (PRE). Te Percent Relative Efficiencies (PREs) of te different estimators are computed wit respect to te adapted estimator using te formula: (, )= 00 for i=, ---------- (5) 5. Empirical Study In tis section, te performance of te suggested estimators ave been analyzed wit respect to te estimators considered in tis paper. To acieve tis, te data set of state wise area, production and productivity of major spices in India was used. In tis data set, te study variable (Y) is productivity in metric tons, te first auxiliary variable (X) is area in tousand ectares, and te second auxiliary variable (Z) is production in tousand tons. From eac stratum states are selected. Te summary of te data is given in te following tables. N n X Y Z C x C y C z x y β β 9 90.534.5 50.3.54 0.745.50 6.7.4.383 9 90.6693.3486 4.95.55 0.78.7 6.3.476 3.564 9 84.956.3434 38.48.443 0.766.669 5.36.584 4.57 β z if Table : Data statistics of parameters Parameters Stratum I Stratum II Stratum III yx.643 0.4 0.89 0 θ θ θ ( yz) ( xz) 5.6047 0 3.9563 0 4.985 0.774 0.78 0 4.4645 0 S yx 50.566 7.5386 60.66 S yz 4.53 0.97 S xz 5758.6 355.74 048.7586 Volume 3 Issue 9, September 04 Paper ID: SEP4658 454
International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 Table 3: Values of parameters ν 0 =3.37859 ν = 8960.84 ν = 64358.93 S x β β * = =4400 S z =4945.833 = 5.4 0 6 β = 3. 0 5 C z =.63 C x =.5 =4.474 9.6 0 β * = 4. 0 β ( x ) = 6.55 ( z) β Table 4: Summary of µ rst µ rst Stratum I Stratum II Stratum III.890 4.77 4.506 µ 300-03.789-7.504 -.6 µ 0-35.34-97.55-94.07 µ 0-0344.88-896.55-8379.3 µ 0 5557.4 5034.483 4034.483 µ 0 487586.07 460689.655 96557.4 µ 030 543793.03 460689.655 35574.38 µ 0 µ 0 06896.55 068965.57 943793.03 µ 003 465574.38 4487586.07 3758606.897 Table 5: PRE of te different estimators wit respect to Estimators s st, y s t s t s reg s pr s pr PRE 4.5 00 4.3458 07.63 3.5684 Table 6: Estimators wit teir MSE values Estimators s st, y s t s reg s pr s pr MSE values 0.40353 0.09 0.437 0.077 0.0673 6. Conclusion Table 5 reveals tat te suggested estimators s pr i, for i=, as te igest PRE among oter estimators considered in tis paper. So tat te suggested estimators in stratified random sampling provides a sufficient Volume 3 Issue 9, September 04 improvement in variance estimation compared to te. s t It is also observed from Table 5 tat te sample and regression estimators are less efficient tan s.table t 6sows tat te proposed estimators of S is more y efficient tan te traditional estimator of population variance of interest variable in stratified random sampling according to te data set of a population considered in tis paper. Teoretically, it as been establised tat, in general, te regression type estimator is more efficient tan te ratiotype estimators. However, in tis paper te regression estimator of is not efficient tan te sample estimator S y and te proposed ratio-type estimators of population variance of interest variable. From te above results and discussion it is observed tat incorporating prior information s obtained from te two auxiliary variables improves population variance of interest variable in stratified random sampling sceme. As a recommendation based on results, te proposed ratio-type variance estimators may be preferred over traditional ratio- type and sample estimator of population variance for te use in practical applications. Tis paper can be improved by adding iger order Taylor series terms. In fortcoming studies, we recommended to develop improved variance estimators by adapting te estimators of Rajes Singi and Mukes Kumar (0). References [] Agarwal, S. K., Two auxiliary variates in ratio metod of estimation. Biometrical Journal,, 569-573 (980). [] Amed, M. S., Raman, M. S. and Hossain, M. I,. Some competitive estimators of finite population variance using multivariate auxiliary information, Information and Management [3] Sciences, (), 49 54, (000). [4] Amed, M. S., Walid, A. D. and Amed, A. O. H. Some estimators for finite population [5] variance under two-pase sampling,statistics in Transition, 6(), 43 50, (003). [6] Al-Jarara, J. and Amed, M. S. Te class of cain estimators for a finite population [7] variance using double sampling, Information and Management Sciences, 3(), 3 8, (00). [8] Arcos, A., et al. Incorporatingte auxiliary information available in variance estimation. Applied Matematics and Computation, 60, 387-399, (005). [9] Cand, L. Some ratio type estimators based on two or more auxiliary variate, P.D. Dissertation, Iowa State university, Ames, IOWA, USA, (975) [0] Das, A. K. and Tripati, T. P., Use of auxiliary information in estimating te finite population variance, Sankya,C, 40, 39-48 (978). [] Gupta, S and Sabbir, J., Variance estimation in simple random sampling using auxiliary information, HacettepeJournal of matematics and Statistics, 37, 57-67 (008). [] H. S. Jajj and G.S. Walia, A Generalized Differencecum-Ratio Type Estimator for te Population Variance Paper ID: SEP4658 455
International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 in Double Sampling, IAENG International Journal of Applied Matematics, 4:4, IJAM_4_4_0 (Advance online publication: 9 November 0). [3] Isaki, C, T., Variance estimation using auxiliary information, Journal of American Statistical Association, 78, 7-3 (983). [4] J. Subramani G. Kumarapandiyan, Estimation of Variance Using Known Coefficient of Variation and Median of an Auxiliary Variable, Journal of Modern Applied Statistical Metods May 03, Vol., No., 58-64 [5] Kadilar, C and Cingi, H, Ratio estimators for te population variance in simple and stratifiedrandom sampling, Applied Matematics and Computations (005a) [6] Kadilar, C. and Cingi, H. A new estimator using two auxiliary variables, Applied Matematics and Computation 6, 90-908, (005b). [7] Kadilar, C. and Cingi, H. A new ratio estimator in stratified random sampling, Communications in Statistics: Teory and Metods 34, 597-60, (005c). [8] Kadilar, C. and Cingi, H. Ratio estimators in stratified random sampling, Biometrical Journal45, 8-5, 003. [9] Kadilar, C. and Cingi, H., Improvement in variance estimation using auxiliary information, Hacettepe Journal of matematics and Statistics, 35, -5 (006a). [0] Kadilar, C. and Cingi, H., Ratio estimators for population variance in simple and Stratified sampling, Applied Matematics and Computation, 73, 047-058 (006b). [] Kendall, M. and Stuart, A. Te Advanced Teory of Statistics: Distribution Teory, (Volume) (Griffin, London, 963). [] Koyuncu. N and Kadilar.C (009)' Family of Estimators of Population Mean Using Two Auxiliary Variables in Stratified Random Sampling', Communications in Statistics - Teory and Metods,38:4,398 47 [3] Liu, T. P, A general unbiased estimator for te variance of a finite population, Sankya [4] C, 36(), 3 3 (974) [5] Murty, M. N., Sampling Teory and Metods, Statistical Publising Society Calcutta, India, (967). Appendixes Volume 3 Issue 9, September 04 [6] Prasad, B. and Sing, H.P. Some improved ratio-type estimators of finite population variance in sample surveys, Communications in Statistics: Teory and Metods 9, 7-39, 990. [7] Prasad B, Sing HP. Unbiased estimators of finite population variance using auxiliary information in sample surveys, Commununication in Statistics: Teory and Metods, Vol. (5): pp.367 376, 99 [8] Rajes Sing and Mukes Kumar, Improved Estimators of Population Mean Using Two Auxiliary Variables in Stratified Random Sampling, Vol.8, 65-7 (0) [9] Sabbir, J. and Yaab, M. Z. Improvement over transformed auxiliary variable in estimating te finite population mean, Biometrical Journal 45, 73-79, 003. [30] Sing, D. and Caudary, F. S., Teory and analysis of sample survey designs, New -Age International Publiser,(986). [3] Sing, S. and Kataria, P. (990), An estimator of finite population variance, Journal of Te Indian Society of Agricultural Statistics, 4(), 86-88. [3] Sing HP and Sing R, Improved ratio-type estimator for variance using auxiliary information, Journal of Te Indian Society of Agricultural Statistics, 54(3): 76 87, (00). [33] Sing, H. P., Tailor, R. and Kakran, M. S., An improved estimator of population mean using power transformation, Journal of te Indian Society of Agricultural Statistics, 58, 330 (004). [34] Srivastava, S. K. and Jajj, H. S., A class of estimators using auxiliary information for estimating finite population variance, Sankya, C, 4, 87-96 (980). [35] Srivastava, S. K. and Jajj, H. S. Classes of estimators of finite population mean and [36] Variance using auxiliary infomation, Journal of Indian Social and Agri. Statist., 47(), 9 8,(995) [37] Subramani, J. and Kumarapandiyan, G., Variance estimation using quartiles and teir functions of an auxiliary variable, International Journal of Statistics and Applications,, 67-7 (0). [38] Upadyaya, L. N. and Sing, H. P., Use of auxiliary information in te estimation of population variance, matematical forum, 4, 33-36 (983) [39] Wolter, K. M. Introduction to Variance Estimation (Springer-V erlag, 985). Appendix A Te MSE of te ratio type variance estimator in te stratified random sampling in te presence of two auxiliary variables can be obtained using te first degree approximation in te Taylor series metod defined by ( ) -------------------------------------------------------------------------- (A.) Were = d d d 3 d 4 d 5 d 6 d 7 d 8 d 9 suc tat = (,,,,,,,, ),,,,,,,, = (,,,,,,,, ),,,,,,,,, = (,,,,,,,, ),,,,,,,, = (,,,,,,,, ),,,,,,,,, = (,,,,,,,, ),,,,,,,, = (,,,,,,,, ),,,,,,,,, Paper ID: SEP4658 456
International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 = (,,,,,,,, ),,,,,,,, = (,,,,,,,, ),,,,,,,,, = (,,,,,,,, ),,,,,,,, and 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 3 34 35 36 37 38 39 3 4 4 43 45 46 47 48 49 4 = 5 5 53 54 5 56 57 58 59 --------------------------. 6 6 63 64 65 6 67 68 69 7 7 73 74 75 76 7 78 79 8 8 83 84 85 86 87 8 89 9 9 93 94 95 96 97 98 9 Here (,,,,,,,, )=,,,,,,,, and is te variance-covariance matrixes of (,,,,,,,, ). Note tat = =, = = and = =. According to equation (A.), we obtain for te estimator, as follows, Let ν 0 = + ( ) ν == + ( ) ν == + ( ), ten we ave = ( ) ( ) ( ) Volume 3 Issue 9, September 04 Paper ID: SEP4658 457 ( ) ( ) ( ) We obtain te MSE of using (A.), as ( ) H + H -------------------------------------------------------------------------- (A.3) Were = k ω = ( Y ) Y H = 4 4 ( ) k ω s y ω Y s y st y = + 4 COV + y, y V st yst k H + ( ) V 4 Y COV y, COV y, V y ( ) (, ) (, ) + (, ) (, ), (, ),, (, ) (, ) (, ), 8 ( )( )(, ) (, ) ( )( ) (, ) + (, ) 8 (, ) (, ) (, ) + (, ) + ( ) + ( ) 8 ( ) 4 ( ) (, ) +4 ( ) (, ) + ( ) + ( )( ) (, ) (, ) + (, ) (, ) + ( ) (, ) + ( ) 4 ( ),, (, ) (, ) (, ) (, ) Appendix B Te MSE of te regression estimator for te population variance in te stratified random sampling in te presence of two auxiliary variables can be obtained using te first degree approximation in te Taylor series metod as follows: = ( ) ( ) ( ) ( ) ( ) ( ) and,using (A.) and (A.),
H 3 Were = H H 3 International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 + -------------------------------------------------------------------------------- (B.) β ω, +4 ( ),, + (, ) (, ) + (, ) (, ) β ω, +4 ( ),, + (, ) (, ) + (, ) (, )+4 ( ) (, ) (, ) (, ) + (, )+ β ω (, ) +8β ( )( )(, ) (, ) + (, ) (, ) +8 ( )( )(, ) (, ) + (, ) (, )+8 β ( )( ) (, ) (, ) (, ) + (, )+ ( )+4 ( ) ( ) (, ) + ( ) + ( )+4 ( ) ( ) (, ) + ( ) Appendix C Te MSE of te proposed estimator and for te population variance in te stratified random sampling in te presence of two auxiliary variables can be obtained using te first degree approximation in te Taylor series metod as follows: = () () ()( () ) ( ) ( ) ( ) ( ) () () () ( ) ( () ) ( ) ( () ) Volume 3 Issue 9, September 04 Paper ID: SEP4658 458 ( () ) (A.) and (A.), we ave () () ()( () ) H + H ---------------------------------------------------- (C.) 4 Were = 4 ( ) (, ) (, ) 4 ( ) (, ) (, ) H 4 () (), 4 ( ) () (, ) (, ( () ) ) (, ) 8 8 ( )( ) () ( )( ) ( () ) 4 ( () ),, (, () ) ( () ) (, ) (, ) (, ) + (, ) (, ) (, ) (, ) + (, ) + ( ), ()( () ) (, ) +4 ( ) (, ) + ( ) + () ( ) 8 ( ) () ( )( ) () () ( ) + ()( () ) (, ) (, ) + (, ) (, ) + ( ) (, ) + ( ) 4 ( ) () (, () ) (, ) (, ( () ) ) (, ) = () () ()( () ) ( ) ( ),, ( ) ( ) () () () ( ) ( () ) ( ) ( () ) Using (A.) and (A.), we ave () () ()( () ) H + H ------------------------------------------------ (C.) 5 Were H = 4 ( ) 5 ( (), ) (, ) 4 ( ) ( ( () ), ) (, ) (), 4 ( ) (),, (, () ) (, ) (, ( () ) ) (, ) ( () ) (, ) 8 ( )( ) ( () ), 8 ( )( ) () (, ) (, ) (, ) + (, ) (, ) (, ) + (, ) + ( () ) () ( ) + Using
4 International Journal of Science and Researc (IJSR) ISSN (Online): 39-7064 Impact Factor (0): 3.358 ( ) () ( )( ) ()( () ) (, ) +4 ( ) (, ) + ( ) + () ( ) 8 ( ) () ()( () ) (, ) (, ) + (, ) (, ) + ( ) (, ) + ( ) 4 ( ) () (, () ) (, ) (, ( () ) ) (, ),, Appendix D = ( ) ( ) ( ), =, = = () =, () =, () =, ( )= - is te population kurtosis of te variate of interest in stratum. ( )= -is te population kurtosis of te first auxiliary variable (X) in stratum. ( )= - is te population kurtosis of te second auxiliary variable (Z) in stratum. = = ( ( ) ) = = 3 = = 4 = ( ) = ( ( ) ) = ( ) = 5 = ( ) = 6 = ( ) = 7 ( ( ) ) = ( ) = 8 = ( ) = 9 = =, = 3 = = 3, = 4 = 4 =, = ( () ), 5 = 5 =, = = 6 6 =, =, 8 = 8, = = 7 = 7 =, = ( () ), 4 = 4 = (, )= = 9 = 9, =, 3 = 3 = (, )= 5 = 5 = (, ) =, 7 = 7 ( =, )= 6 = 6 = 35 = =( 53, )= (, )= = 8 8 = (, ) =, 9 = 9 = 38 = = ( 83, )= = 34 43 = (, )=, 36 = 63 = (, )= = 37 = ( 73, )=, 39 = 93 = (, )= = 45 54 = (, )=, 46 = 64 = (, )= = 47 = 74 (, ) = ( () ), 48 = 84 = (, )= = 49 = (, 94 )=, 56 = 65 = (, )= = 57 = (, 75 )= = 58 = (, ) = 85 = 59 = 95 = 68 = (, )=(, )= 86 = 67 = (, 76 )=, 69 = 96 = (, )= = 78 = (, 87 )=, 79 = 97 = (, )= = 89 = (, )= 98 Volume 3 Issue 9, September 04 Paper ID: SEP4658 459