American Journal of Mathematics and Statistics 06, 6(): 8-5 DOI: 0.59/j.ajms.06060.0 A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme Öge Akkuş Muğla Sıtkı oçman Universit, Faculit of Science, Department of Statistics, Menteşe, Muğla, Turke Abstract There are man ratio and product tpe mean estimators proposed in different sampling schemes in sampling literature. However, a large number of them are more efficient than the others in certain circumstances, depending on the structure of the data used. Determining the theoretical efficienc conditions of the estimators significantl facilitates the work of practitioners and provides appropriate help in identifing the most suitable estimators according to the structure of the data set the used. In this stud, the performance of man ratio and product tpe mean estimators is compared over their Bias, the Mean Square Error values and specific efficienc conditions are determined. In addition, a discussion is held on which of the mean estimator achieves to obtain the best estimation over a real data set on the table olive production taken from the records of the Turkish Statistical Institute for 00. ewords sampling Auiliar variable, Olive production, Ratio estimator, Population mean, Product estimator, Two phase. Introduction In the sampling theor, including the population information of the auiliar variable X, highl correlated with the stud variable Y, significantl reduces the estimation errors. However, population information of the auiliar variable (s) is not ver eas to obtain. In such cases, the use of two-phase sampling (or double sampling) is required for the parameter estimation. In the first phase, population information of the auiliar variable X is estimated using the second auiliar variable which is highl correlated with X. In the second stage, variables X, Y and are recorded from the sub-sample drawn from the primar sample to obtain the most efficient estimates. In the current literature, there are numerous mean estimators proposed in the two phase sampling scheme. The efficienc levels of these estimators are proved theoreticall but under certain conditions. In order to use the theoretical findings effectivel, characteristics of the studied data sets should be well identified and the most appropriate estimators for the data structure should be selected. The purpose of this stud is to reveal the efficienc conditions of the proposed ratio and product tpe estimators, theoreticall b making pairwise * orresponding author: oge.akkus@mu.edu.tr (Öge Akkuş) Published online at http://journal.sapub.org/ajms opright 06 Scientific & Academic Publishing. All Rights Reserved comparisons. For this purpose, Bias and Mean Square Error () values of all the estimators are calculated. Additionall, a discussion is held on the determination of the best and the worst mean estimators over a real data set on the table olive production taken from the records of the Statistical Institute of Turke for 00. In section, the two phase sampling method is briefl introduced and the classical ratio and product mean estimators in Simple Random Sampling (SRS) plan are presented. The list of estimators is given with their Bias and values in Section. Section 4 and 5 include the theoretical comparison of the estimators and a real data application, respectivel. Finall, all the obtained findings are summaried in Section 6.. Two Phase Sampling Let us consider that the mean of a population consisting of N units is estimated. Stud variable Y can be eplained better with the use of auiliar variable X. However, in some studies, population information of X cannot be obtained. In this case two phase sampling is used. The purpose of this method in the first stage is to estimate the population information of X, most effectivel and to achieve the most efficient estimates of the population parameters of Y with selected sub-sample in the second phase of the sampling method. When the population information of X cannot be reached, a primar sample with n unit n N is drawn from the population with N units b the method of
American Journal of Mathematics and Statistics 06, 6(): 8-5 9 Simple Random Sampling (SRS) without replacement. In some cases, information relating to the variable X can be obtained using the second auiliar variable. In this case, variables X and are observed in the primar sample. After estimating the necessar information relating to the variable X in the first stage, it is passed to the second stage. The population mean of Y in the second stage is estimated with the help of the information of X obtained in the first stage. The sample drawn from the primar sample with the method of SRS without replacement is called sub-sample and denoted b n n n [0], [5]... lassical Ratio Estimator While making estimations related to the interested variable of Y, adding the information of the auiliar variable X, which is highl correlated with Y, can reduce the estimation errors, significantl. If the relation between variables Y and X is linear and the proportion of i does i not change from sample unit to sample unit, ratio estimators should be used. Let us draw from a sample with n units from a population with N units with the method of SRS and let us assume that, measurements can be obtained related to the sample i i units. Also assume that all the conditions required for the use of the ratio tpe estimators are provided. In this case, the proposed classical estimator is defined as follows []. r r X () Here, and denotes the sample means of the variables X and Y, respectivel, and X is the population mean of X. Bias and values given b Eq.() and Eq.() of the classical ratio estimator in the method of SRS is obtained b using the difference method. r Bias Yf () r Y f f In Eq.(), f n is the sampling proportion S N Y and S represent standard deviations of Y and X, respectivel X S Y Y and S X () X denote variation coefficient of Y and X is defined for the simplicit in equations and calculations [9]... lassical Product Estimator Let us assume that we draw a sample with n units from a population with N units b the method of SRS. If the stud variable Y is inversel proportional to the auiliar variable X, in other words, if i i multiplication does not change much from sampling unit to sampling unit, the use of the product tpe estimators in the estimation of the population information of Y is suggested. The classical product estimator, p, in SRS is as the following [] p (4) X Here, and denotes the sample means of the variables X and Y, respectivel, and X is the population mean of X. Bias and values of the classical product estimator in SRS is defined as the following [6]. p Bias Yf (5) p Y f f Researchers have suggested man new or modified ratio and product tpe mean estimators in different sampling designs b inspiring the classical ratio and product estimators in SRS. The acceptabilit of these new proposed estimators requires the theoretical proof related to the high efficienc level than previousl suggested estimators. However, each new proposed estimator cannot be more efficient than previousl proposed estimators in all situations. Therefore, it is required to demonstrate conditions in which the are more efficient than others, theoreticall. In the following section, some of the proposed two-phase sampling estimators are listed in chronological order along with their Bias and values.. Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme The list of man ratio and product tpe mean estimators in the current literature is presented in Table. (6) Table. Ratio and Product Tpe Mean Estimators, Bias and Values in Two Phase Sampling Estimator Name Bias and lassical Ratio Bias Y Y
0 Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme lassical Product Bias Y Y opt Srivastava (970) Bias Y Bias Y min Y min Y 4 (a b 0) hand (975) 4 Bias Y 4 Y 5 6 b 7 b b 8 opt (a b ) k 9 k opt hand (975) iregera (980) Upadhaa (990) Upadhaa (995) Upadhaa (00) 5 Bias Y 5 Y Bias( ) Y 6 6 Y 7 Bias Y 7 Y 8 Bias Y Bias Y min 8 8 Y Y min 8 Bias 9 Y k k Bias =Y min 9 9 kk k k Y Y min 9
American Journal of Mathematics and Statistics 06, 6(): 8-5 0 (a b ) opt Upadhaa and Singh (00) 0 Bias Y Biasmin 0 Y 0 Y min 0 Y opt (a b ) opt opt 4 4 4opt (a b ) Upadhaa and Singh (00) Upadhaa and Singh (00) Upadhaa and Singh 4 (00) Singh (00) Bias Y Bias Y min Y min Y Bias Y min Bias Y Y min Y Bias Y min Bias Y Y min Y 4 4 Bias Y 4 4 4 Y Y min 4
Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme 5 () 5 () 5opt () (a () b ) () 6 () 6 () 6opt (a () b ) () () 7 a 7 7opt * opt opt a a a 8 a 0 * * a b * * a b * * a b * * a b opt * * * a a b. R / Singh (00) Singh (00) Singh Famil of (00) Singh, Upadhaa and handra Famil of (004) 5 5 Bias Y 5 5 5 Y Y min 5 6 6 Bias Y 6 6 6 Y Y min 6 7 7 Bias Y 7 7 7 Y Y min 7 Bias 8 8 Y Y S R min 8.
American Journal of Mathematics and Statistics 06, 6(): 8-5 9 (a b ) 8 8opt 4 5 0 4opt 5opt 8 opt k k k ( ) 6 4 6 i (i 8,0,,4,5,6) a 0 6opt a a b Singh hauhan and Sawan (007) Singh, hauhan and Sawan Famil of (007-a) Espejo (007) Singh, hauhan, Sawan and Smarandac he Famil (008) Bias 9 Y ( ) 8 ( ) 9 8 8 Y min 9 Y Bias 0 0 4 4 Y 5 58 ( ) Y 4 4 5 8 58 min 0 Y Bias Y k Y k k Y min Bias 0 Y 6 9 6 6 min Y k k k opt houdhur and Singh (0) Bias k ( k ) Y k ( k ) k Y k Y min Definations of some statements in Table are given below.
4 Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme n N : orrection factor for the sub-sample : orrection factor for the primar-sample. n N n n : Difference between the correlation factors of sub- and primar samples. : orrelation coefficient of population between variables Y and X. S Y S : oefficient of variation of variable X. : Abbreviation made for the simplicit. Y : oefficient of variation of variable Y b X X : Regression coefficient between the auiliar variables X and. : Estimation of the regression coefficient : Skewness of variable : urtosis of variable : Standard deviations of variable. 4. Theoretical Efficienc omparison of the Estimators The theoretical efficienc comparisons of all the estimators over values in Table are presented in Table. Table. Pairwise Efficienc omparisons of the Estimators Over Values Estimator Efficienc Level Estimator ondition 0 lassical Product Srivastava Espejo 0 0 hand- lassical Ratio Estimator hand- 4 iregera Upadhaa 0.. Upadhaa Upadhaa and Singh----4 Singh--- Singh Famil 0
American Journal of Mathematics and Statistics 06, 6(): 8-5 5 Same Efficienc Level Singh, hauhan, Sawan Singh, hauhan, Sawan and Smarandache Famil Upadhaa- min 9 Singh, Upadhaa and handra Famil min 8 for a * =, b * =, Singh, hauhan and Sawan famil of estimators ( ) 0 houdhur and Singh estimator ( ) ( ) for 0 Srivastava Espejo hand- ( ) 0 4 0 hand- 0 lassical Product Estimator iregera Upadhaa 4 0. Upadhaa Upadhaa and Singh----4 Singh--- Singh Famil 4 0 Singh, hauhan and Sawan Singh, hauhan, Sawan and
6 Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme Smarandache Famil Upadhaa- 4 0 Singh, Upadhaa and handra Famil min 8 for a * =, b * = and Singh, hauhan and Sawan famil. ( ) 0 houdhur and Singh ( ) ( ) for 0 hand- hand- iregera Upadhaa 0 Srivastava and Espejo Estimators Upadhaa Upadhaa and Singh----4 Singh--- Singh Famil Singh, hauhan, Sawan and Smarandache Famil Singh, hauhan and Sawan 0 Upadhaa- Singh, Upadhaa and handra Famil min 8
American Journal of Mathematics and Statistics 06, 6(): 8-5 7 houdhur and Singh Singh, hauhan and Sawan famil of estimators For a * =, b * = and 0. 0 Same Efficienc Espejo min min hand- 0. iregera Upadhaa 0 0 Upadhaa Upadhaa Upadhaa and Singh----4 hand- Estimator Singh--- Singh Famil Singh, hauhan, Sawan 0 Singh, hauhan, Sawan and Smarandache Famil Singh, Upadhaa and handra Famil 4 min 8 for a * =, b * = and Singh, hauhan and Sawan famil of estimators 0 ( ) 0 houdhur and Singh ( ) ( )
8 Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme iregera 4 0 Upadhaa. Upadhaa Upadhaa and Singh----4 Singh--- Singh Famil Singh, hauhan and Sawan 0 hand- Estimator Singh, hauhan, Sawan and Smarandache Upadhaa 4 0 Singh, Upadhaa and handra Famil 5 min 8 for a * =, b * = and Singh, hauhan and Sawan famil houdhur and Singh ( ) 0 and ( ) ( ) iregera Estimator Upadhaa Upadhaa Upadhaa and Singh----4 Singh--- Singh Famil Singh, hauhan and Sawan Singh, hauhan, Sawan and Smarandache 0 0 Upadhaa- 0
American Journal of Mathematics and Statistics 06, 6(): 8-5 9 Singh, Upadhaa and handra Famil 6 min 8 for a * =, b * = and Singh, hauhan and Sawan famil of estimators ( ) ( ) 0 houdhur and Singh ( ( ) ( ) ( ) ) Upadhaa Upadhaa and Singh----4 Singh--- Singh Famil Singh, hauhan and Sawan Singh, hauhan, Sawan and Smarandache Upadhaa Estimator Upadhaa- 0 and Singh, Upadhaa and handra Famil 7 min 8 for a * =, b * = and Singh, hauhan and Sawan famil of estimators 0 ( ) 0 houdhur and Singh and ( ) ( ) Upadhaa Estimator- Upadhaa and Singh----4 Singh--- Singh Famil Singh, hauhan and Sawan 0
0 Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme Singh, hauhan, Sawan and Smarandache famil of estimators Singh, Upadhaa and handra Famil 9 min 8 for a * =, b * = and Singh, hauhan and Sawan 0 Espejo 0 houdhur and Singh ( ) 0 and ( ) ( ) Upadhaa- 0 Upadhaa Upadhaa and Singh ---4 Singh -- Singh Famil of Singh hauhan and Sawan Singh-hauhan-Sa wan and Smarandache Famil Singh-Upadhaa- handra Famil of Estimator Singh, Upadhaa and handra Famil hauhan and Sawan a * =, b * = and famil 0 Espejo houdhur and Singh Singh, hauhan and Sawan famil houdhur and for 0 and ( ) a * =, b * = and Singh min 8 Singh-hauhan and Sawan Famil of Estimators houdhur and Singh 0 and ( ) 5. Application In this part of the stud, a real data application is presented to illustrate how we determine the most suitable estimator according to the data structure among numerous proposed mean estimators in the two phase sampling scheme. Additionall, it is aimed to investigate the validit of the theoretical findings in practice and to obtain the most efficient estimator of Y. To this end, the table olive production data obtained from 87 districts taken from the records of the Turkish Statistical Institute for 00 is used. Definitions of the stud variable of Y and auiliar variables
American Journal of Mathematics and Statistics 06, 6(): 8-5 of X and are given below. Y: The amount of produced table olives (tons) X: The number of fruit trees in that age : ollective areas of fruit (Decar) The average number of trees in the age of the fruit (X) is estimated using the information of the collective areas of fruit (). In this stud, in which the population unit is districts, primar and sub samples are drawn b the method of SRS without replacement. When the sampling tolerance value is determined as 00, d=00, the sie of the primar sample drawn from the population with N=87 units b SRS is n 65. Similarl, the sie of the sub-sample drawn from the primar sample using SRS at the same tolerance level is found 06.470. Estimations related to the variables of X and are made in the selected primar sample whereas Y, X and for the sub-sample determined. The population and sample information for the table olive production data is given in Table. Table. Descriptive Statistics of the Population, Primar Sample and Sub Sample for the Table Olive Production Data POPULATION Mean Y=06.60 X 408.969 767.69 Variance S Y=8776.96 S X=896869458.8 S =79997.7 oefficient of Variation urtosis Skewness Y =.60989999 X =.5007888 =.594508 (Y)=49.966575 (X)=0.4004 ()=59.508056 (Y)=5.9476 (X)=9.5406958 ()=.8967 Slope YX 0.005457 Y 0.08958 X 5.565464 orrelation oefficient YX 0.80455 N 87 PRIMARY SAMPLE Y 0.887998 X 0.96504 Mean =04. =9675.99 =59.7555 Variance S Y =59.05559 S X =6095.8866 S =9.4858 oefficient of Variation Y =.54489 X =.698575 =.480877 (Y )=.9497448 (X )=6.5677868 urtosis (Y )=4.787 (X )=5.46666844 Skewness ( )=9.097654 ( )=4.09005504 Slope YX 0.00696 Y 0.7947 X 7.580964 orrelation oefficient n' 65 ryx 0.644769 r 0.70754958 Y X SUB SAMPLE r 0.895068 Mean =09.5477 =6646.7585 =494.490566 Variance s =07.4879 s =5085.559 s =746.98 oefficient of Variation c =.95056076 c =.090778 c =.77077547 urtosis Skewness ()=8.7848998 ()=7.44865 ()=8.0455 ()=.7679598 ()=.779056 ()=.68755090 Slope b 0.04599 b 0.79557 b 4.4759799
Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme orrelation oefficient r 0.7654069 r 0.664557 r 0.794765887 n 06 0.00595 0.58 opt 0.58 a 0.00576 =0.560 opt 0.44 b.5 0.007 =0.964 opt 0.94 opt.00 kopt 0.969. 489 opt 0.966 opt opt 0.560 kopt 0.759. 745 4opt 0.58 R. 0.674 kopt 0.7608 0. 9676 5opt 6opt 79.748 opt 0.994 opt opt opt 4opt 5opt 6opt 7opt 8opt 0.560 The same directional correlation is observed both between variables Y and X and Y and. Therefore, it would be right to use the ratio estimators in the estimation of average amount of the table olive production. Low efficienc level related to the product tpe mean estimators is epected. The estimated values, Bias and values for the table olive production data are given in Table. The last column of Table includes the efficienc order of all the estimators according to the values. Table 4. Efficienc Order of the Estimators for the Olive Production Data Estimator Estimated Value Bias (Theoretical) (Theoretical) (Order) 54.7 6.6 499.068 9 lassical Ratio lassical Product 7.60 8.4 964.9 0 6. 6.779 870.457 5 Srivastava 4 965.045 45.95 4650.767 8 hand -
American Journal of Mathematics and Statistics 06, 6(): 8-5 5 549.94 -.9 087.989 hand- 6 b iregera 7 b b 9.09 8.647 4464.87 7 704.50.957 87.06 4 Upadhaa 8 964.777 6.6 464.479 6 Upadhaa 9 k Upadhaa- 0 467.784-4.6 499.068 9 965.04 4.65 464.479 6 Upadhaa and Singh- 96.67 4.65 464.479 6 Upadhaa and Singh- 69.098 4.65 464.479 6 Upadhaa and Singh- 67.060 4.65 464.479 6 Upadhaa and Singh-4 4 595.0 4.65 464.479 6 Singh- (00)
4 Öge Akkuş: A Review on the Theoretical and Empirical Efficienc omparisons of Some Ratio and Product Tpe Mean Estimators in Two Phase Sampling Scheme () 5 () 95.50 4.65 464.479 6 Singh- (00) () 6 () 85.998 4.65 464.479 6 Singh- (00) 7 a a Singh Famil (00) a b a b 8 a b a b 4.550 -.0488 705.050 Singh, Upadhaa and handra Famil (004) 9 964.97 6.6 464.479 6 Singh, hauhan and Sawan 4 5 0 48.87.975 7795.868 Singh, hauhan and Sawan Famil k k.479 6.68 870.46 5 Espejo (007) ( ) 6 4 6 i (i 8,0,,4,5,6,7,9) (i 4 ) 6 4 ( 6) 4 740.06 0 464.48 6 Singh, hauhan, Sawan and Smarandache Famil of (008) k k 66.56 46.97 7797.74 houdhur and Singh (0)
American Journal of Mathematics and Statistics 06, 6(): 8-5 5 The best estimator of the average olive production is Singh, Upadhaa and handra Famil of Estimator ) in parallel with the theoretical findings according to ( 8 the criterion whereas hand Estimator- ( 5 ) gives the worst estimation of all. 6. Results In this stud, man ratio and product tpe mean estimators in the eisting literature are eamined and the efficienc conditions are revealed, theoreticall. Additionall, the validit of the theoretical findings in practice is investigated over a real data set on the table olive production in 00 in Turke. Efficienc orders are determined according to the values. The best estimator of the average olive production is Singh, Upadhaa and handra Famil of Estimator ( 8 ) whereas hand Estimator- ( 5 ) gives the worst estimation of all. Another important point to be emphasied is that the efficienc level of the classical product estimator is much lower than the classical ratio estimator both in theor and practice. Namel, that the relationship between variables Y and X is positive, high level of efficienc of the ratio estimators is an epected result. When the kurtosis and the coefficient of variation of the second auiliar variable are added to the estimator, rising efficienc level is observed. When the correlation coefficient between variables Y and the first auiliar variable X is known, Singh, hauhan and Sawan Famil of Estimator ( 0 ) gives more efficient estimations. In the light of the results of this stud, applied researchers can get pre information on which estimator(s) will give better estimation results according to the nature and characteristics of the data set the used. Additionall, researchers that make theoretical studies in this field can also find an opportunit to assess the estimators and efficienc levels together in the two phase sampling literature. REFERENES [] hand, L., 975. Some Ratio-Tpe Estimators Based on Two or Auiliar Variables. Ph.D. Dissertation, Iowa State Universit, Ames, Iowa. [] houdhur, S., Singh, B.., 0, A lass of hain Ratio-Product Tpe Estimators with Two Auiliar Variables Under Double Sampling Scheme, Journal of the orean Statistical Societ, 4(): 47-56. [] ochran,., 977, Sampling Techniques, John Wile & Sons, New York. [4] iregera, B.., 980. A hain Ratio Tpe Estimator in Finite Population Double Sampling Using Two Auiliar Variables. Metrika, 7: 7-. [5] Ögül N., 007, İki Safhalı Örneklemede Ortalama Tahmin Edicileri, MhD Dissertation, Hacettepe Universit, Ankara, Turke. [6] Singh, G.N., Upadhaa, L.N., 995. A lass of Modified hain Tpe Estimators Using Two Auiliar Variables in Two Phase Sampling, Metron, 5: 7-5. [7] Singh, G.N., 00. On The Use of Transformed Auiliar Variable in the Estimation of Population Mean in Two Phase Sampling. Statistics in Transition, 5, (): 405-46. [8] Singh, G.N., Upadhaa, L.N., 00. An Empirical Stud of Modified Ratio Estimators in Two Phase Sampling in Presence of oefficient of Variation of the Auiliar Variable, Statistics in Transition, 5(): 9-6. [9] Singh, H.P., Upadhaa, L.N., handra, P., 004, A General Famil of Estimators for Estimating Population Mean Using Two Auiliar Variables In Two Phase Sampling, Statistics in Transition, 6(7): 055-077. [0] Singh, H.P., Singh, S., im, J., 006. General Families of hain Ratio Tpe Estimators of The Population Mean with nown oefficient of Variation of The Second Auiliar Variable in Two Phase Sampling. Journal of the orean Statistical Societ, 5(4): 77-95. [] Singh, R., hauhan, P., Sawan, N., 007. Ratio Estimators in Simple Random Sampling Using Information on Auiliar Attribute. Auiliar Information and A Priori Value in onstruction of Improved Estimators, Renaissance High Press: 7-7. [] Singh, R., hauhan, P., Sawan, N., 007a. A Famil of Estimators for Estimating Population Mean Using nown orrelation oefficient in Two Phase Sampling. Statistics in Transition-new series, 8(): 89-96. [] Singh, H.P., Espejo, M.R., 007. Double Sampling Ratio-Product Estimator of a Finite Population Mean in Sample Surves, Journal of Applied Statistics, (): 7-85. [4] Singh, R., hauhan, P., Sawan, N., Smarandache, F., 008, Improvement in Estimating Population Mean Using Two Auiliar Variables in Two Phase Sampling, http://ariv.org/ftp/ariv/papers/080/080.094.pdf. [5] Srivastava, S.., 970. A Two Phase Sampling Estimator in Sample Surves. Austrilian Journal of Statistics, : -7. [6] Sukhatme, P. V., Sukhatme, B. V., Sukhatme, S., 984. Sampling Theor of Surves with Applications. Indian Societ of Agricultural Statistics, New Delhi, India, and The Iowa State Universit Press, Ames, Iowa, USA, 56p. [7] Upadhaa, L.N. and Singh, G.N., 00, hain tpe estimators using transformed auilar variable in two-phase sampling. Advances in Modeling and Analsis, 8(-): -0.