A Comparison of Exponential Type Mean Estimators in Two Phase Sampling

Transcription

1 American Journal of Mathematics and tatistics 7, 7(5: 9-5 DOI:.593/j.ajms A omparison of Eponential Type Mean Estimators in Two Phase ampling Öge Akkuş Department of tatistics, Muğla ıtkı Koçman University, Muğla, Turkey Abstract In this study, eponential type mean estimators in two phase sampling method are introduced. Bias and mean square error equations are derived. All the studied estimators are listed chronologically. In order to compare the efficiencies of the estimators, data on table olive production taken from Turkish tatistical Institute from 87 regions of Turkey in are used. Mean value, mean square error values and relative efficiency values are calculated. The best and the worst estimators are determined. Keywords Two Phase ampling, Eponential Estimator, Ratio Estimator, Product Estimator, Relative Efficiency, Bias, Mean quare Error. Introduction When we want to estimate the population mean of units, the study variable can be eplained better by using the auiliary variable X. However, in some studies, the population information of auiliary variable X may not be accessible. In this case, two phase sampling method is used. The aim of this method is to estimate the population information of the auiliary variable X in the first phase in the most efficient way and reach the most efficient estimates of the population parameters of variable via selected sub-sample in the second phase. If population information of the auiliary variable X cannot be reached, a sample is selected in order to estimate the information of variable X in the first phase of the sampling method which takes place in two phases. In the first phase, the sample selected with imple Random ampling (R without replacement is called the primary sample and is composed of n ( n< units. In some cases, information on variable X can be obtained using a second auiliary variable (Z. In this case, X and Z variables are observed in the primary sample. In the first phase, the necessary information for variable X is estimated and passed to the second phase. In the second phase, the population mean of the variable is estimated with the aid of the information of the auiliary variable X obtained in the first phase. For this purpose, the sample drawn from primary sample with R without replacement is called a sub-sample n< n ([], [7]. and consists of n units ( orresponding author: oge.akkus@mu.edu.tr (Öge Akkuş Published online at opyright 7 cientific & Academic Publishing. All Rights Reserved The most important point to note in sample selection is sample sie. Decisions about how large the sie of both samples should have a significant impact on the efficiencies of estimates. The sub-sample sie should not eceed the primary sample sie. The larger the primary sample, the greater the efficiencies of the predictions. Therefore, the sie of the samples, sampling method and estimators to be used are very important in terms of the efficiency and accuracy of the estimates both in the selection of primary and sub-samples. In this study, the primary and sub-samples are selected with R without replacement. In two phase sampling method, there are ratio, product, regression and eponential types estimators. The ratio estimator is used when the correlation between the study variable and the auiliary variable X is directly proportional; product estimator when it is inversely proportional and the regression estimator is used when it can be obtained from a different point from the origin. When there is a curvilinear relationship between variables, it is observed that using eponential estimators gives more efficient results []. In this study, eponential type mean estimators in two phase sampling method are investigated. The equations of Bias and Mean quare Error (ME of all estimators that are subject to the study and the theoretical proofs are derived. In the application part of the study, the table olive production data taken from Turkish tatistical Institute obtained from 87 regions of Muğla province of are used. Bias, ME and Relative Efficiency (RE values are calculated. Estimators are compared with each other and the most efficient and worst estimator for the data set used is reached.

2 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling. Literature Review The first classical ratio and product estimator in the two phase sampling method was proposed by [8]. [5] proposed a new ratio and product estimator by adapting their proposed ratio and product estimators in R to the two phase sampling method. [] proposed new ratio and product type eponential estimators in two phase sampling method, inspired by the ratio and product type eponential estimators proposed by [] in R. [7] proposed a new estimator by adapting their estimator to two phase sampling method in the form of a combination of ratio and product estimators proposed by [] in R. Inspired by ratio and product type eponential estimators proposed by [] in R, [3] proposed a new eponential estimator class by adapting their proposed estimator class in R to the two phase sampling plan. [6] proposed two new estimators, adapting the estimator they proposed in R to two phase sampling method. [3] proposed new ratio-eponential and product-eponential estimators in two phase sampling method, inspired by the estimators of [8] and [4]. [9] proposed a chain-type ratio and chain type product eponential estimators using second variable information in two phase sampling method following the ratio and product type eponential estimators proposed by []. [6] proposed ratio-cum-product and chain-type ratio eponential estimators using two auiliary variable information. [] proposed a chain-ratio type eponential family of estimator based on the ratio and product estimators using two auiliary variables in estimating the population mean proposed by [] in two phase sampling method. [] proposed a class of chain-ratio and product type eponential estimators in the form of a combination of their proposed chain ratio and chain product type eponential estimators in two phase sampling. [5] proposed new chain-ratio and chain-product type eponential estimators based on their proposed chain-ratio and chain-product type eponential estimators in two phase sampling. [] proposed three new chain-type eponential estimators inspired by the estimators of [] in R and [] in two phase sampling. [9] proposed a class of chain-ratio type eponential estimator for the estimation of population mean using two-auiliary-variable information in two phase sampling. [] proposed a class of chain-ratio-product type eponential estimator for the estimation of population mean using two-auiliary-variable information in two phase sampling. [] proposed a new class of chain-ratio-product type eponential estimator in estimating population mean using two-auiliary-variable-information in two phase sampling. Inspired by [] and [3] estimators, [4] proposed chainratio and chain-product type eponential estimators in the case of two auiliary variables in two phase sampling for the estimation of the population mean. 3. Eponential Type Mean Estimators in Two Phase ampling The list of eponential type mean estimators in two phase sampling in current literature, Mean quare Error (ME and Bias equations are presented in Table. Definitions of some statements in Table are given below. Here, the sub-sample and primary sample correction factors ( λ and λ are defined, respectively as follows. λ n (3. n n λ (3. n λ3 λ λ (3.3 orrelation coefficient of population between and X, and Z, and X and Z are defined as the following. ( i ( Xi X y i y ρ y (3.4 y y ( Xi X ( i i i ( i ( Zi Z y y Z Z i ρ y ( i i i ( i ( Xi X( Zi Z y y i ρ ( Xi X ( Zi Z i i (3.5 (3.6 Here, the population coefficient of variation for variables, X and Z are given in Eq.(3.7, Eq.(3.8 and Eq.(3.9, respectively. y (3.7 y (3.8 X (3.9 Z

3 American Journal of Mathematics and tatistics 7, 7(5: 9-5 American Journal of Mathematics and tatistics 7, 7(5: 9-5 Table. The hronological Order of the Eponential Type Mean Estimators in Two Phase ampling y y r ukhatme lassical Ratio Estimator (96 ( r λ 3 ( y ( r λ y + λ 3 ( y Bias y K ME y K yp y Re Md y ep + P e Md y ep + 3 t wt W w ii i ( t y, Bias t d t y ep, Bias t + 8 4K t y ep, Bias t λ K ( λ ( d d 3 y ( ( d d 3 y y y ep ( ep Ky + opt ukhatme lassical Product Estimator (96 ingh and Vishwakarma Ratio-Eponential Estimator (7 ingh and Vishwakarma Product-Eponential Estimator (7 ingh, hauhan, awan and marandache Eponential Estimator (7 ingh, huhan and awan Eponential Estimator (8 ( p λ 3 y ( p λ y + λ 3 ( + y Bias y K ME y K 3 ( Re Md λ 3 y 8 ( λ λ Bias K ME Re Md y + 3 K y 4 ( P e Md λ 3 + y 8 ( λ + λ + Bias K ME P e Md y 3 K y 4 + Bias ( tw λ w w 3 w ρ yy 8 w ME ( tw λ y + λ 3 w K y 4 ( ( ME t λ λρ min w y 3 y Bias ( y λ 3 8K y 8 Biasmin ( y λ 3 K y 8 ME ( y λ y + λ3 K y ( ( ME y λ λρ min y 3 y

4 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Table. (continued twd wd y+ wd trsd + wd trsed a + b trsd y a + b ( a + b ( a + b trse y ep ( a + b + ( a + b ax w θ ax + b ; d wd wd + ; θρ y y Ky wd ( opt θ θ tqd qd y+ qd tpsd + qdtpsed Qd a + b tpsd y a + b ( a + b ( a + b t psed y ep ( a + b + ( a + b ax θ ax + b qd qd qd + ; q + q K d d y ( K q y d opt θ E Re d cy ep + λ3 A λρ 3 y y 4 B λ λρ + c y 3 y y opt A B ingh, Kumar and marandache Eponential Estimator (8 ingh, Kumar and marandache Eponential Estimator (8 adav, ingh, Tiwari, Kumar and hukla Ratio-Eponential Estimator ( 3 wd Bias ( twd θ λ3 wd + wd θρ yyλ 3 wd + 8 ( ( + wd λ y λ3θ d θλ3ρ y y d ( ( min wd y 3 y ME t w w ME t λ λρ qd Bias ( tqd λ3 θqd K y θ 8 ( ( + qd λ y λθ 3 d + θ d ρy yλ 3 ( ( + ME t q q ME t λ ρ λ min qd y y 3 ( 3 K y Bias E Re d ( c + c λ ( E Red ( + λ y + λ3 ρy y λ 3 ( ME c c c c c c 4 ( A MEmin E Re d 4B

5 American Journal of Mathematics and tatistics 7, 7(5: American Journal of Mathematics and tatistics 7, 7(5: Table. (continued EP e d cy ep + λ A y y B + λ c y 3 y y opt A cb Z Re y ep Z + Z Pe y ep Z + adav, ingh, Tiwari, Kumar and hukla Product-Eponential Estimator ( ingh and houdhury Ratio-Eponential Estimator ( ingh and houdhury Product-Eponential Estimator ( ( ( ( EP e d + A min ( EP e d 3 K y Bias EP e d ( c + c λ ME c B ca ME 4B 3 ( Re ( λ3 + λ ( λ3 y + λ y 8 ( λ + ( λ + λ λ λ Bias K K ME Re y 3 3K y K y 4 ( P e ( λ3 + λ + ( λ3 y + λ y Bias K K 8 ME ( Pe λ + y ( λ3 + λ + λ3k y + λ K y 4 oor-ul-amin and Hanif Z X t4 yep Ratio-Product Eponential Z + X + Estimator ( oor-ul-amin and Hanif X Z Ratio-hain Type Eponential t5 yep + X + Z + Estimator ( ( ( az + b a + b t red y ep ( ( + + az b + a b θ K θ y adav, Upadhyaya, ingh and hatterjee Ratio-hain Type Family of Eponential Estimator (3 λ3 λ Bias( t4 ( 3 4ρyy ( + ρ 4ρ yy 8 8 λ λ ME( t 3 ( y 4ρy y ( + 4ρy y ρ 4 4 ( λ ( + 3 λ 3 Bias t5 3 ρyy ρ ρ yy 4 λ λ ME( t 3 ( y 4ρyy ( 4ρyy + ρ Bias( tred λ3 + θ λ λρ 3 y y λρ y y 8 8 ( θλ ME tred λy + λ3 + ρy y λ3 θλρ y y 4 4 ME ( min tred λ y + λ3 ρyyλ3 λρ yy 4

6 4 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling 4 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Table. (continued Z Z + Re y ep ( ep Z Z + + ( λρ 3 y y y y ( λ + λ3 + opt a + b + a b t y ep a + b + + a b θ K θ y a + b + a b t y ep a + b + + a b θ K θ y a + b a + b t y ep a + b + a + b y y Ky ρ ; y θ ρy Ky θ ingh, houdhury and Kalita Ratio-hain Type Family of Eponential Estimator (3 Kalita, ingh and houdhury Ratio-hain Type Family of Eponential Estimator- (3 Kalita, ingh and houdhury Product-hain Type Family of Eponential Estimator- (3 Kalita, ingh and houdhury Ratio-hain Type Family of Eponential Estimator- (3 λ3 λ λ ρ y y + ρ y y λ3ρ y y 8 Bias ( RPe λ λ3 λρ y y + + λy + λ + λ3 λ3 λ3ρ y y 4 4 ME( RPe + λ3ρ y y + λ λ y y λρ y y + λ3 ( λρ 3 y y y ( y ME min RPe λ y ( λ + λ3 Bias( t 3 λ3 + 3 θ λ λρ 3 y y λρ y y + λρ 8 8 θ λ ME( t λy + λ3 + ρy y λ3 θλρ y y 4 4 MEmin ( t λ y + λ3 ρyyλ3 λρ yy 4 θ Bias( t λ3 θ λ 3 y y y y 8 8 θ λ ME( t λy + λ3 + + ρy y λ3 + θλρ y y 4 4 MEmin ( t λ y + λ3 + ρyyλ3 λρ yy 4 3 θ Bias( t + λ3 + θ λ λ3 ρy y λρ y y θλ ME( t λy + λ3 + λρ 3 y y θλρ y y 4 4 ( ( λ + λ λ ME t K K min y 3 y y

7 American Journal of Mathematics and tatistics 7, 7(5: American Journal of Mathematics and tatistics 7, 7(5: Table. (continued a + b a + b t y ep a + b + a + b y ρy K y ; y y K y θ ρ θ Kalita, ingh and houdhury Product-hain Type Family of Eponential Estimator- (3 5 3 λ 3 + θ λ ρ + θ λ Bias ( t θ + λ3 ρy y y y θλ ME( t λy + λ3 + 3 y y + θλρ y y 4 4 ( ( λ λ λ ME t K K min y 3 y y y Z T ep Z + ingh and Majhi hain Type Eponential Estimator- (4 BiasT ( 3 λ + λ3 λρ 3 y y λρ y y 8 λ ME( T + λ3 + λy λρ y y λρ 3 y y 4 Z ingh and Majhi T3 y ep hain Type Eponential Estimator- + (4 ( ( Z + β + β V y ep ( ( Z β β δ Ky Vishwakarma and Gangele hain Type Ratio-Eponential Estimator (4 BiasT ( 3 λ + 3 λ3 λρ 3 y y λρ y y 8 λ 3 ME( T3 λ + + λy λρ y y λρ 3 y y 4 ( 3 3 δλρ y y λρ 3 y y Bias V λ3 + δ λ δλ ( λ 3 δ λ ME V λy + + δλρ y y λρ 3 y y 4 4 MEmin ( V λ y + λ3 λρ yy λ3λρ yy 4

8 6 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling 6 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Table. (continued ( Z + β k ep ( ( Z β β T ( ( Z β + β + + ( k ep ( ( Z β β ρ y y δ ; ( k ρ yy k + Z Z yrpe y ( + Z + + Z ( + + ( + ( λ3 + λ λ ρ λ λρ 3 y y y y opt Z y ep Re Z + Z y ep Pe Z + Vishwakarma, Kumar and Gangele hain Type Ratio-Product Eponential Family of Estimators (4 Vishwakarma and Kumar hain Type Ratio-Product Eponential Family of Estimators (4 ingh and Ahmed hain Type Ratio-Eponential Estimator (5 ingh and Ahmed hain Type Product-Eponential Estimator (5 ( 4k ( ( k λ3 δ λ ( λρ 3 y y δλρ y y BiasT ( ( k ( ( ( ME( T λy + λ3 + δ λ k λρ 3 y y + δλρ y y 4 ( ( ME T λ λρ λρ min y 3 y y ( 4 ( ( λ λ ( λρ λρ Bias( yrpe y y + y y 8 ( ( ( ( ME( yrpe λy + λ3 + λ λρ 3 y y y y 4 ( λρ + 3 y y λρ y ( y ME min yrpe λ y λ + 3 λ ( λρ y y + λλρ y y + ( λ 3λ + λ 4 3 Bias ( Re ( λρ 5λλρ y y 5λ + 6 ME( λy + ( λ3 + λ + ( λρ 3 y y Re y y 6 ( λρ y y y y y y + ( λ 3λ 3λ 4 3 Bias ( Pe + ( 3 λρ λρ λ 6 ME( λy + λ3 + λ 3 y y Pe y y 6 6

9 American Journal of Mathematics and tatistics 7, 7(5: Population coefficient of variation between and X, and Z, and X and Z are given by Eq.(3., Eq.(3. and Eq.(3., respectively. The sample variances s, y follows. y ρ (3. X y y y y ρ (3. Z y y y ρ (3. XZ s and s are defined as n y ( i (3.3 n- i s y -y n ( i (3.4 n- i s - The population variances n ( i (3.5 n- i s -, and by Eq.(3.6, Eq.(3.7 and Eq.(3.8, respectively. X Z are given ( i (3.6 -i - X ( i (3.7 -i X -X Z ( i (3.8 -i Z -Z In addition, the following definitions are used for convenience in equations. K K K y y y ρ y δ Z Z + β y ax ρ y θ + ax ρ 4. Application az θ az + b In this part of the study, to compare the efficiencies of eponential type mean estimators proposed in two phase sampling in the literature, data on table olive production taken from Turkish tatistical Institute in 87 regions of Muğla province in are used. The definitions of the study variable and auiliary variables X and Z are given below [4]. X: Amount of table olive produced (Ton : umber of fruit trees Z: Area of bulk fruits (Decar The average number of trees in the fruiting age ( X is estimated using the knowledge of the area of the collective fruit trees. The main aim of this study is to estimate the average table olive production ( in the most efficient way, where the population unit is regions. The primary and sub-samples are drawn with the R method without replacement. The primary sample sie to be drawn from the population ( 87 by the R method when the amount of error that can be tolerated is d 3, ( t y n d 3 n n' n If the amount of error that can be tolerated for the sub-sample sie to be selected from the primary sample with n 65 is taken as d 3, the sub-sample sie to be selected with the R method is, ( t y n d 3 n n n For the population and selected sample, the values in the following tables are observed. The values of Bias, ME and RE of all eponential mean estimators subject to the study and theoretical proofs are calculated on the basis of the these values. The population information obtained for table olive production, the primary sampling data and the sub-sample information is given in Table, Table 3 and Table 4, respectively. Table 5 shows the variables used when estimating Bias, ME and RE values of the proposed estimators. In order to compare the efficiency of the eponential mean estimators proposed in the two phase sampling method, Bias, ME and RE values calculated based on the average table olive production are given in Table 6. From these estimators, in which the ME value is the lowest estimator is the most efficient estimator.

10 8 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Table. Population Information Related to the Table Olive Production Data Mean 36.6 X Z Variance X Z tandard Deviation X Z tandard Error X Z oefficient of Variation. 6 X 3. 5 Z oefficient of Kurtosis β (.9637 β ( X.4 β ( 49 oefficient of kewness β ( 94 β ( X 354 ( Z β Z.38 lope β X.55 β Z.896 β XZ orrelation oefficient ρ X.845 ρ Z.888 ρ XZ Table 3. Primary ample Information Related to the Table Olive Production Data Mean y Variance y tandard Deviation y tandard Error y oefficient of Variation y oefficient of Kurtosis β ( y.949 β (.568 β ( 3 36 oefficient of kewness β ( y.38 β (.4666 β ( 4 lope y orrelation oefficient y n b.63 b.379 b 7.58 y r.634 r.775 r.895 y Table 4. ub-ample Information Related to the Table Olive Production Data Mean y Variance y tandard Deviation y tandard Error y oefficient of Variation y oefficient of Kurtosis β ( y.7849 β (.4439 β ( 8 7 oefficient of kewness β ( y 7679 β ( 779 β ( lope by. 5 by. 7 b orrelation oefficient ry ry. 664 r n 6

11 American Journal of Mathematics and tatistics 7, 7(5: Table 5. ome Parameter Values Related to the Proposed Eponential Estimators λ.59 λ.6 λ 3.34 a b 3.5 K y. 58 K y. 56 opt.5 β opt.48 θ opt.99 w. 66 ingh, hauhan, awan and marandache Estimator (7 w.5 ingh, Kumar and marandache Estimator (8 q.564 ingh, Kumar and marandache Estimator (8 c c opt opt adav, ingh, Tiwari, Kumar and hukla Estimator ( k Vishwakarma, Kumar and Gangele Estimator (4 w. 566 ingh, hauhan, awan and marandache Estimator (7 w.48 ingh, Kumar and marandache Estimator (8 q.6667 ingh, Kumar and marandache Estimator (8 U Kalita, ingh and houndhury Estimator (3 w.4 ingh, hauhan, awan and marandache Estimator (7 w ingh, Kumar and marandache Estimator (8 A A.9596 adav, ingh, Tiwari, Kumar and hukla Estimator ( u 89. Kalita, ingh and houndhury Estimator (3 K. 965 w. 63 ingh, hauhan, awan and marandache Estimator (7.4 ingh, hauhan and awan Estimator (8 q.6467 ingh, Kumar and marandache Estimator (8 B B.5837 adav, ingh, Tiwari, Kumar and hukla Estimator ( δ.99 Vishwakarma and Gangele Estimator (4 Eponential mean estimators are listed in chronological order in the two phase sampling in Table 6. When the table is eamined, it is observed that most of the proposed estimators give more efficient results than the classical ratio and product estimators. When the efficiencies of the estimators are compared based on the ME values, the most efficient estimator for estimating the average table olive production is "Kalita, ingh and houndhury Ratio Estimator-" with ME value; the second most efficient estimator is "adav, Upadayaya, ingh and hatterer Estimator"; while the worst estimator is the "ukhatme lassical Product Estimator" with ME value. The RE- values in Table 6 are calculated according to the classical ratio estimator. When these values are eamined, it is observed that "Kalita, ingh and houndhury Ratio Estimator-" is.4 times efficient than the classical ratio estimator; adav, Upadayaya, ingh and hatterer Estimator is.77 times higher than the classical ratio estimator and "ukhatme lassic Product Estimator" is.5 times more efficient than the classical ratio estimator. RE- values are calculated according to the classical product estimator. When these values are eamined, it is revealed that "Kalita, ingh and houndhury Ratio Estimator-" is times and "adav, Upadayaya, ingh and hatterer Estimator" is observed to be. times more efficient than the classical product estimator. RE-3 values are calculated according to the mean-per-unit estimator ( y. When these values are eamined, it is found that "Kalita, ingh and houndhury Proportional Estimator-" is.57 times; "adav, Upadayaya, ingh and hatterter Estimator" is.9 times and "ukhatme lassic Product Estimator" is.6 times more efficient than mean-per-unit estimator.

12 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Table 6. Bias, ME and RE Values of ome Proposed Eponential Mean Estimators in Two Phase ampling Estimator Estimated Value Bias ME RE- RE- RE-3 ME Order t y Basit Tahmin Edici y y r ukhatme lassical Ratio Est. (96 yp y ukhatme lassical Product Est. (96 Re Md y ep + ingh and Wishwakarma Ratio Est. (7 P e Md y ep + ingh and Wishwakarma Product Est. (7 tw wy + wy ep + wy ep + + ingh, hauhan, awan and marandache Est. (7 y y ep ( ep ingh, hauhan and awan Est. (8 twd wd y+ wd trsd + wd trsed ingh, Kumar and marandache Ratio Est. (8 tqd qd y+ qd tpsd + qdtpsed ingh, Kumar and marandache Est. ( E Re d cy ep + adav, ingh, Tiwari, Kumar and hukla Ratio Est. (

13 American Journal of Mathematics and tatistics 7, 7(5: 9-5 American Journal of Mathematics and tatistics 7, 7(5: 9-5 Table 6. (continued EP e d cy ep + adav, ingh, Tiwari, Kumar and hukla Product Est. ( Z Re y ep Z + ingh and houndhury Ratio Est. ( Z Pe y ep Z + ingh and houndhury Product Est. ( Z X t4 yep Z + X + oor-ul-amin and Hanif Ratio Est. ( Z X t5 yep + Z + X + oor-ul-amin and Hanif Product Est. ( ( ( az + b a + b t red y ep ( ( az b + + a + b adav, Upadayaya, ingh and hatterter Est. (3 Z Z Re y ep βep + Z Z + + ingh, houdhury and Kalita Est.(

14 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling Table 6. (continued a + b a + b t y ep a b + a + b + Kalita, ingh and houndhury Ratio Est.- (3 a + b a + b t y ep a + b + a + b Kalita, ingh and houndhury Product Est.- (3 a + b a + b t y ep a + b + a + b Kalita, ingh and houndhury Ratio Est.- (3 a + b a + b t y ep a + b + a + b Kalita, ingh and houndhury Product Est. (3 y Z T ep Z + ingh and Majhi Est. (4 Z T3 y ep + ingh and Majhi Est. (

15 American Journal of Mathematics and tatistics 7, 7(5: American Journal of Mathematics and tatistics 7, 7(5: Table 6. (continued ( ( Z + β + β V y ep ( ( Z β β Vishwakarma and Gangele Est. (4 ( ( ( Z β Z β + β + + T k ep ( ( + ( k ep Z β ( ( Z β β + β Vishwakarma, Kumar and Gangele Est.(4 Z Z yrpe y ( + Z + + Z Vishwakarma and Kumar Est. (4 Z y ep Re Z + ingh and Ahmed Ratio Est. (5 Z y ep Pe Z + ingh and Ahmed Product Est. (

16 4 Öge Akkuş: A omparison of Eponential Type Mean Estimators in Two Phase ampling 5. Results In this study, eponential type mean estimators in two phase sampling method are investigated. The estimators proposed at various times in the literature search are the subject of the study, the theoretical proofs are made, and the Estimator, Bias and ME equations are derived. In order to determine the efficiencies of the proposed estimators, data on table olive production taken from Turkish tatistical Institute from the 87 regions of Muğla province in are used. Estimated value, Bias, ME and RE values of the estimators are calculated. When these values are eamined, it is seen that "Kalita, ingh and houndhury Ratio Estimator-" has the lowest ME value ( and therefore the most efficient estimator among two phase eponential mean estimators in the study. The second most efficient estimator is observed to be "adav, Upadayaya, ingh and hatterter Estimator" with ME value. It is also observed that the estimator that predicts the variable with the least efficiently is the "ukhatme lassical Product Estimator" with ME value. In the net step of this study, it is planned to propose new estimators or families of estimators that yield better results than the estimators available in the literature in the two phase sampling. AKOWLEDGEMET We are thankful to cientific Research Unit of Muğla ıtkı Koçman University for their financial support and for providing necessary guidance concerning projects implementation. REFEREE [] Akkuş, Ö. & Günay,. (6. İstatistiksel tahmin teorisi, Gai Kitabevi, Ankara, IB: , 34s. [] Bahl,., & Tuteja, R. (99. Ratio and product type eponential estimators. Journal of Information & Optimiation ciences, (, [3] hand, L. (975. ome ratio type estimator based on two or more auiliary variables. Unpublished Ph. D. dissertatiton. Lowa tate University, Ames, Lowa. [4] Gandge,.., Varghese, T., & Prabhu-Ajgaonkar,. G. (993. A note on modified product estimator. Pakistan Journal of tatisticc, 9(3B, [5] Kalita, D., ingh, B. K., & houdhury,. (3. Improved eponential chain ratio and product-type estimators for finite population mean in double sampling. International Journal of tatistical ciences, 3, -37. [6] oor-ul-amin, M., & Hanif, M. (. ome eponential estimators in survey sampling. Pakistan Journal of tatistics, 8(3, [7] Ögül,. (7. İki safhalı örneklemede ortalama tahmin edicileri (yüksek lisans tei, Hacettepe Üniversitesi, Ankara. [8] Prasad, P. (989. ome improved ratio type estimators of population mean and ratio in finite population sample surveys. ommunications in tatistics: Theory and Methods, 8, [9] ingh, B. K., & houdhury,. (. Eponential chain ratio and product type estimators for finite population mean under double sampling scheme. Global Journal of cience Frontier Research Mathemmatics and Decision cience, (6, 3-4. [] ingh, B. K., houdhury,., & Kalita, D. (3. A class of eponential chain ratio-product type estimator with two auiliary variables under double sampling scheme. Electronic Journal of Applied tatistical Analysis, 6(, [] ingh, G.., & Majhi, D. (4. ome chain-type eponential estimators of population mean in two-phase sampling. tatistics in Transition, 5(, -3. [] ingh, H. P., & Viswakarma, G. K. (7. Modified eponential ratio and product estimators for finite population mean in double sampling. Austrian Journal of tatistics, 36(3, 7-5. [3] ingh,., hauhan, P., & awan,. (8. On linear combination of ratio and product type eponential estimator for estimating the finite population mean. tatistics in Transition-ew eries, 9(, 5-5. [4] ingh, R. K., & Ahmed, A. (5. Improved eponential ratio and product type estimators for finite population mean under double sampling scheme. International Journal of cientific and Engineering Research, 6(4, [5] ingh, R., hauhan, P., awan,., & marandache, F. (7. Ratio-product type eponential estimator for estimating finite population mean using information on auiliary attribute. R. ingh, P. hauhan,. awan, F. marandache, R. ingh, P. hauhan,. awan, & F. marandache (Dü içinde, Auiliary Information and A Priori Value in onstruction of Improved Estimators (s UA, 74s: Renaissance High Press. [6] ingh, R., Kumar, M., & marandache, F. (8. Almost unbiased estimator for estimating population mean using known value of some population parameter(s. Pakistan Journal of tatistics and Operation Research, 4(, [7] ingh, R; hauhan, P; awan, ; marandache, F. (7(b. Almost unbiased eponential estimator for the finite population mean. R. ingh, P. hauhan,. awan, & F. marandache, Auiliary Information and A Priori Value in onstruction of Improved Estimators, (s Renaissance High Press, UA, 74p. [8] ukhatme, B. V. (96. ome ratio-type estimators in two-phase sampling. Journal of the American tatistical Association, 57, [9] Vishwakarma, G. K., & Gangele, R. K. (4. A lass of chain ratio-product type eponential estimators in double sampling using two auliary variates. Applied Mathematics and omputation, 7, 7-75.

17 American Journal of Mathematics and tatistics 7, 7(5: [] Vishwakarma, G. K., Kumar, M., & Gangele, R. K. (4. A general class of chain ratio-product type eponential estimators in double sampling using two auiliary variates. The Philippine tatistician, 63(, 9-. [] Vishwakarma, G., & Kumar, M. (4. An improved class of chain ratio-product type estimators in two-phase sampling using two auiliary variables. Journal of Probability and tatistics, 4, -6. [3] adav,. K., ingh, L., Tiwari, W., & hukla, A. K. (. Efficient eponential ratio and product type estimators of population mean under double sampling. International Journal of omputing, (, 93-. [4] [] adav, R., Upadhyaya, L.., ingh, H. P., & hatterjee,. (3. A chain ratio eponential type estimator in two phase sampling using auiliary information. tatistica, 73(, -34.