Journal of Modern Applied Statistical Methods

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1 Journal of Modern Applied Statistical Methods Volume 3 Issue Article Two Parameter Modified Ratio Estimators with Two Auxiliar Variables for Estimation of Finite Population Mean with Known Skewness, Kurtosis and Correlation Coefficient Jambulingam Subramani Pondicherr Universit, Puducherr, India, drjsubramani@ahoo.co.in G Prabavath Pondicherr Universit, Puducherr, India, praba.gopal.3@gmail.com Follow this and additional works at: Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theor Commons Recommended Citation Subramani, Jambulingam and Prabavath, G (04) "Two Parameter Modified Ratio Estimators with Two Auxiliar Variables for Estimation of Finite Population Mean with Known Skewness, Kurtosis and Correlation Coefficient," Journal of Modern Applied Statistical Methods: Vol. 3 : Iss., Article 3. DOI: 0.37/jmasm/ Available at: This Regular Article is brought to ou for free and open access b the Open Access Journals at DigitalCommons@WaneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods b an authorized editor of DigitalCommons@WaneState.

2 Journal of Modern Applied Statistical Methods Ma 04, Vol. 3, No., 99-. Copright 04 JMASM, Inc. ISSN Two Parameter Modified Ratio Estimators with Two Auxiliar Variables for Estimation of Finite Population Mean with Known Skewness, Kurtosis and Correlation Coefficient Jambulingam Subramani Pondicherr Universit Puducherr, India G. Prabavath Pondicherr Universit Puducherr, India Consider the two parameter modified ratio estimators for the estimation of finite population mean using the skewness, kurtosis and correlation coefficient of two auxiliar variables. The efficiencies of the proposed modified ratio estimators are assessed with that of the simple random sampling without replacement (SRSWOR) sample mean and some of the existing ratio estimators in terms of mean squared errors. The entire above is explained with the help of certain natural populations available in the literature. Kewords: Mean squared error; natural populations; percentage relative efficienc; simple random sampling Introduction In surve sampling, consider the problem of estimating the population mean N i i for a finite population U = {U,U,,UN} of N distinct and N identifiable units, where the value i is measured on Ui, i =,,3,,N. Normall the population mean is estimated b the sample mean obtained from a random sample of size n drawn b simple random sampling without replacement (SRSWOR) from a finite population, when there is no auxiliar information available. Suppose that there is an auxiliar variable X available that is positivel correlated with a stud variable, in this case, either a ratio estimator or linear regression estimator ma be used to improve the efficienc of the SRSWOR Dr. Subramani is an Associate Professor in the Department of Statistics. him at: drjsubramani@ahoo.co.in. G. Prabavath is a Ph.D. Scholar in the Department of Statistics. her at: praba.gopal.3@gmail.com. 99

3 TWO PARAMETER MODIFIED RATIO ESTIMATORS sample mean under certain conditions (see, Cochran (977) and Murth (97) for example). Further improvements can be achieved on the ratio estimator b using known parameters such as skewness, kurtosis, quartiles and coefficient of variation of the auxiliar variable; the resulting estimators are called modified ratio estimators. For further details on the modified ratio estimators, readers are referred to Kadilar and Cingi (004, 009), Singh and Tailor (003, 005), Singh (003), Sisodia and Dwivedi (98), Subramani (03), Subramani and Kumarapandian (0a, b, c, 03), Upadhaa and Singh (999), and an and Tian (00). If two auxiliar variables exist, then several modified ratio estimators have been proposed b linking together ratio estimators, product estimators and regression estimators in order to obtain more efficient estimators. For more detailed discussion about ratio estimators and their modifications using two auxiliar variables readers are referred to: Abu-Daeh et al. (003), Bandopadha (980), Cochran (940), Kadilar and Cingi (004, 005), Khare et al. (03), Murth (97), Naik and Gupta (99), Olkin (958), Perri (004, 007), Rao and Mudholkar (97), Raj (95), Sahoo and Swain (980), Singh (003), Singh (95, 97), Singh and Tailor (003, 005), Srivenkataramana (980), Srivenkataramana and Trac (98), Tailor et al. (0), and Trac et al. (99). Existing Estimators with and without auxiliar variables If (,,,n) is a random sample of size n drawn from a population of size N using SRSWOR, then the population mean can be estimated b the sample mean n i i, which is an unbiased estimator, and its variance is given b: n V ( f ) n, f. () n ( N ) N N ( ) = S, where S ( ) i i The ratio estimator for estimating the population mean of the stud variable is defined as X RX. () R x The mean squared error of the ratio estimator approximation is: R to the first degree of 00

4 SUBRAMANI & PRABAVATH ( f ) MSE( R ) ( C CX pxcxc ). (3) n Singh (003) suggested a ratio estimator with two auxiliar variables for estimating a population mean: X X x x ( )( ). (4) The mean squared error of to the first order of approximation is: ( f ) MSE( ) ( C CX C ) X p x C x C px x C x C x (5) n Singh and Tailor (005) suggested the following modified ratio cum product estimator with known correlation coefficient between auxiliar variables: X p x p. () x x x x x px x X px x The mean squared error of to the first order of approximation is: where f * * * * * MSE( ) C Cx K x Cx ( K x Kx ) x n (7) C C C X X K K K x * * x x, x x, x, x and Cx C x C x X x x X x x and xx is the coefficient of correlation between X and X. Kadilar and Cingi (005) proposed a new ratio estimator using two auxiliar variables as: 0

5 TWO PARAMETER MODIFIED RATIO ESTIMATORS X X 3 b ( X x ) b ( X x). (8) x x The mean squared error of 3 to the first order of approximation is: S R B S x R B S x f MSE 3 R B S x n R B S x R B R B S xx (9) S S where B, B, R and R. x x Sx S x X X Perri (007) suggested some modified ratio cum product estimators using two auxiliar variables for estimating the population mean: t X X X t X, and, (0) 4 5 t X t t t X where t x ( X x ) and t x ( X x ). The mean squared errors of 4, 5, to the first order of approximation are: MSE MSE MSE f 4 S x x x x x x n f 5 S x x x x xx n f S x x x x x x n () () (3) 0

6 SUBRAMANI & PRABAVATH where R R S, RS, x x x x ( ) RS. and ( ) RS. x x x x x x RS x x This article is concerned with estimating the population mean of a stud variable b two parameter modified ratio estimators with known correlation coefficient, skewness and kurtosis of two auxiliar variables X and X. Proposed Two Parameter Modified Ratio Estimators Whenever one or two auxiliar variables exist, a number of estimators including ratio, regression, product and chain ratio tpe estimators and their linear combinations have been proposed in the literature. These estimators are improved b using the known values of parameters such as skewness, kurtosis and coefficient of variation of the auxiliar variables. All of these estimators are functions of the ratio, product, regression estimators and their linear combinations; hence, an attempt is made herein to introduce the weighted average of the ratio estimators whenever there are two auxiliar variables available. As a result, two parameter modified ratio estimators with known correlation coefficient, skewness, kurtosis and their linear combinations of two auxiliar variables are proposed. When the coefficient of kurtosis ( X) of the auxiliar variable X, and β(x) of the auxiliar variable X is known, the following two parameter modified ratio estimator is proposed: SP X ( X) X ( X ). (4) x ( X) x ( X ) Using the linear combinations of coefficient of kurtosis ( X) of the auxiliar variable X, β(x) of the auxiliar variable X and correlation coefficient between X and X, the following two parameter modified ratio estimators xx are proposed: and SP x x X ( X) x x X ( X ) x x x ( X) x x x ( X ) (5) 03

7 TWO PARAMETER MODIFIED RATIO ESTIMATORS ( X) X x x ( X ) X x x. () ( X) x x x ( X ) x x x Using the linear combinations of coefficient of skewness ( X) of the auxiliar variable X, β(x) of the auxiliar variable X, coefficient of kurtosis β (X) of the auxiliar variable X and ( X ) of the auxiliar variable X the following two parameter modified ratio estimators are proposed: and ( X) X ( X) ( X ) X ( X ) ( X) x ( X) ( X ) x ( X ) (7) ( X) X ( X) ( X ) X ( X ). (8) ( X) x ( X) ( X ) x ( X ) In general, the estimators proposed in (4) to (8) can be defined as particular cases of the estimator: X T X T. (9) x T x T For suitable choices of T and T in (9), the estimators defined in (4) to (8) are obtained. Suppose that, i. if T ( X) and T ( X) in (9), then becomes SP as defined in (4); ( X) ( X ) ii. if T and T in (9), then becomes SP as xx xx defined in (5); xx xx iii. if T and T ( X ) ( X ) defined in (); in (9), then becomes as 04

8 SUBRAMANI & PRABAVATH ( X) ( X ) iv. if T and T ( X ) ( X ) in (9), then becomes as defined in (7); and ( X) ( X ) v. if T and T in (9), then becomes as ( X ) ( X ) defined in (8). Derivation of Mean Squared Error of the proposed estimators The mean squared error of the proposed estimator is derived as follows. If x X x X e0, e, and e, then ( e0 ), x X( e ), X X and x X( e ). From the definition of e 0 and E e0 E e 0 is ( f ) f f obtained where E e0 C, E e Cx, n E e Cx, n n f E( e0e) x C, Cx n f f E( e0e ) x C and ( Cx E e e ) x x C x C x. n n e, The proposed estimator can be written in terms of e 0, e and e as: e [ X T ] [ X T ] ( 0) [ X( e ) T ] [ X ( e ) T ] X X T T ( e0 ) X X T T Xe X e X X ( e ), and 0 e e [ X T ] [ X T ] [ X T ] [ X T ] ( e ) e e 0 ( e ) e e e e 0 ( e ) e e e e e e 0 05

9 TWO PARAMETER MODIFIED RATIO ESTIMATORS Neglecting higher order terms e e e e e e e e e and squaring and taking expectations on both sides results in: MSE( ) E( ) E( e e e ) 0 MSE( ) E e e e e e e e e e MSE( ) E( e ) E( e ) E( e ) E( e e ) E( e e ) E( e e ) f MSE C C C C C C C C C n (0) ( ) x x x x x x x x x x The proposed modified ratio estimator can be easil generalized to include several auxiliar variables. If X, X,,Xk are k auxiliar variables that are positivel correlated with a stud variable, then the generalized modified ratio estimator is defined as G [ X T ] [ X T ] [ X T ]... [ X T ] k k k [ X T ] [ X T ] 3[ X3 T3 ]... k[ X k Tk ] where,,, k are the weights and the T, T,, T k are the known parameters of the auxiliar variables. Efficienc Comparisons The efficiencies of the proposed estimators for estimating the finite population mean are assessed with that of SRSWOR sample mean and other existing estimators, as previousl proposed. From expressions (0) and (), the proposed estimators are more efficient than the SRSWOR sample mean r. The derived conditions are: 0

10 SUBRAMANI & PRABAVATH MSE if V ( ) r C C C C C C C C () x x x x x x xx x x From expressions (0) and (5), the proposed estimators are more efficient than the existing ratio estimator. The derived conditions are: MSE if MSE ( ) C C ( ) C C ( ) C C ( ) C C x x x x x x x x x x () From expressions (0) and (7), the proposed estimators are more efficient than the existing ratio estimator. The derived conditions are: MSE if MSE ( ) * * Cx Cx * * * * {( ) C C ( ) C C ( ) C C } x x x x xx x x (3) From expressions (0) and (9), the proposed estimators are more efficient than the existing ratio estimator 3. The derived conditions are: MSE if MSE 3 ( ) R R S R R S B S B S x x x x R ( S x Sx ) R ( R B )( R B ) Sx x (4) 07

11 TWO PARAMETER MODIFIED RATIO ESTIMATORS From expressions (0) and (), the proposed estimators are more efficient than the existing ratio estimator 4. The derived conditions are: MSE if MSE 4 ( ) R ( ) R Sx R ( ) R Sx { S x R ( ) R S x R ( ) R S xx R R R ( )( ) } (5) From expressions (0) and (), the proposed estimators are more efficient than the existing ratio estimator 5. The derived conditions are: MSE if MSE 5 ( ) R ( ) R Sx R ( ) R Sx { S x R ( ) R S x R ( ) R S xx R R R ( )( ) } () From expressions (0) and (3), the proposed estimators are more efficient than the existing ratio estimator. The derived conditions are: MSE if MSE ( ) R ( ) R Sx R ( ) R Sx { S x R ( ) R S x R ( ) R S xx R R R ( )( ) } (7) 08

12 SUBRAMANI & PRABAVATH where R ( X T ) ( X T ) Numerical Stud The performance of the proposed two parameter modified ratio estimators have been compared with that of the SRSWOR sample mean and some existing modified ratio estimators algebraicall. However, the proposed estimators perform well compared to the existing estimators onl under certain conditions and - for numerical comparisons - the are assessed for certain natural populations. In this connection, two natural populations were considered to assess the performance of the proposed estimators with that of existing estimators. Population is from Singh and Chaudhar (98, p. 77) and population is from Kadilar and Cingi (009, p. 7). The description of the stud and auxiliar variables for the two populations are shown in Table. Table. Description of the stud variable and auxiliar variable Population Stud Variable Auxiliar Variable X Auxiliar Variable X Area under wheat in 974 Area under wheat in97 Area under wheat in973 Length of the fish Length of the head Length of the fin The population parameters and constants computed for the two populations are given in Tables

13 TWO PARAMETER MODIFIED RATIO ESTIMATORS Table. Parameters and Constants of the Populations Parameter N n X X x x xx Pop Pop Parameter S Pop Pop C S x S x C x C x Table 3. Variance/Mean squared error of the existing and proposed estimators for Population Existing Estimators r Proposed Estimators SP SP

14 SUBRAMANI & PRABAVATH Table 4. Variance/Mean squared error of the existing and proposed estimators for Population Existing Estimators r Proposed Estimators SP SP From the values in Tables 3 and 4, the mean squared error of the proposed modified ratio estimators SPj, j,,3,4,5 are less than the variance of SRSWOR sample mean, the mean squared error of the existing modified ratio estimators j; j,,3,...,. Further, to show the efficienc of the proposed estimators, the percentage relative efficiencies (PRE s) of the proposed estimators with reect to the existing estimators is computed b: PRE SPj MSE(.) *00. MSE SPj

15 TWO PARAMETER MODIFIED RATIO ESTIMATORS Table 5. PRE of the proposed estimator SPj for Population α α Proposed Estimators SRSWOR r Existing Estimators Modified Ratio Estimators SP SP SP SP SP SP SP SP

16 SUBRAMANI & PRABAVATH Table 5, continued α α Proposed Estimators SRSWOR r Existing Estimators Modified Ratio Estimators SP SP SP SP SP SP SP SP

17 TWO PARAMETER MODIFIED RATIO ESTIMATORS Table 5, continued α α Proposed Estimators SRSWOR r Existing Estimators Modified Ratio Estimators SP SP SP SP SP SP

18 SUBRAMANI & PRABAVATH Table 5 shows the following ranges for the PRE of the proposed estimators: from 0.0 to in comparison with the SRSWOR sample mean; from 4.99 to 5. in comparison with the existing estimator defined in (4); from 0.94 to.3 in comparison with the existing estimator defined in (); from.30 to 8.9 in comparison with the existing estimator 3 defined in (8); from 4.55 to 87.7 in comparison with the existing estimator 4, 5, defined in (0). Based on these comparisons, it is concluded that the proposed estimators perform better than the SRSWOR sample mean and other existing ratio estimators for the natural population considered in this stud. Table. PRE of the proposed estimator SPj for Population α α Proposed Estimators SRSWOR r Existing Estimators Modified Ratio Estimators SP SP

19 TWO PARAMETER MODIFIED RATIO ESTIMATORS Table, continued α α Proposed Estimators SRSWOR r Existing Estimators Modified Ratio Estimators SP SP SP SP SP SP SP SP

20 SUBRAMANI & PRABAVATH Table, continued α α Proposed Estimators SRSWOR r Existing Estimators Modified Ratio Estimators SP SP SP SP SP SP SP SP

21 TWO PARAMETER MODIFIED RATIO ESTIMATORS Table, continued α α Proposed Estimators SRSWOR r Existing Estimators Modified Ratio Estimators SP SP SP SP Table shows the following ranges for the PRE of the proposed estimators: from 3.94 to in comparison with SRSWOR sample mean; from to in comparison with the existing estimator defined in (4) and defined in (); from 3. to in comparison with the existing estimator 3 defined in (8); from to 88.7 in comparison with the existing estimator 4 defined in (0); from to 8.75 in comparison with the existing estimator 5 defined in (0); from to 88.7 in comparison with the existing estimator defined in (0). 8

22 SUBRAMANI & PRABAVATH Based on these comparisons, it ma be concluded that the proposed estimators perform better than the SRSWOR sample mean and other existing ratio estimators for the natural population considered in this stud. Conclusion This article proposed two parameter modified ratio estimators with known correlation coefficient, skewness and kurtosis of the auxiliar variables and their linear combinations. The mean squared errors of the proposed estimators were derived and compared with that of SRSWOR sample mean, the classical ratio estimator and the existing modified ratio estimators. The performance of the proposed estimators was also assessed with that of the existing estimators for certain natural populations. It was observed from the numerical comparisons that the mean squared errors of the proposed estimators are less than the mean squared error of the existing estimators. Further it was shown that the PREs of the proposed estimators, with reect to existing estimators, range from 0.0 to Hence, the proposed modified ratio estimators are strongl recommended and ma be preferred over existing estimators for practical applications. Acknowledgements The authors wish to extend their gratitude and thanks for the financial assistance received through UGC-Major Research Project, and to the Editor and the Referees for their effort to improve the presentation of the paper. References Abu-Daeh, W. A., Ahmed, M. S., Ahmed, R. A., & Muttlak, H. A. (003). Some estimators of finite population mean using auxiliar information. Applied Mathematics and Computation, 39: Bandopadha, S. (980). Improved ratio and product estimators. Sankhā C, 4: Cochran, W. G. (940). The estimation of the ields of cereal experiments b sampling for the ratio of grain to total produce. Journal of Agriculture Science, 37:

23 TWO PARAMETER MODIFIED RATIO ESTIMATORS Cochran, W. G. (977). Sampling techniques. New ork, N: John Wile & Sons. Kadilar, C., & Cingi, H. (004). Estimator of a population mean using two auxiliar variables in simple random sampling. International Mathematical Journal, 5: Kadilar, C., & Cingi, H. (005). A new estimator using two auxiliar variables. Applied Mathematics and Computation,, Kadilar, C., & Cingi, H. (009). Advances in sampling theor-ratio method of estimation. Oak Park, IL: Bentham Science Publishers. Khare, B. B., Srivastava, U., & Kumar, K. (03). A generalized chain ratio in regression estimator for population mean using two auxiliar characters in sample surve. Journal of Scientific Research, 57: Murth, M. N. (97). Sampling Theor and Methods. Calcutta, India: Statistical Publishing Societ. Naik, V. D., & Gupta, P. C. (99). A general class of estimators for estimating population mean using auxiliar information. Metrika, 38: -7. Olkin, I. (958). Multivariate ratio estimation for finite populations. Biometrika, 45: Perri, P. F. (004). Alcune considerazioni sull'efficienza degli stimatori rapporto-cum-prodotto. Statistica & Applicazioni, (): Perri, P. F. (007). Improved ratio-cum-product tpe estimators. Statistics in Transition-NS, 8(): 5-9. Raj, D. (95). On a method of using multi-auxiliar information in sample surves. Journal of the American Statistical Association, 0: Rao, P. S. R. S., & Mudholkar, G. S. (97). Generalized multivariate estimator for the mean of finite populations. Journal of the American Statistical Association, : Sahoo, L. N., & Swain, A. K. P. C. (980). Unbiased ratio-cum-product estimator. Sankhā C, 4: 5-. Singh, D., & Chaudhar, F. S. (98). Theor and analsis of sample surve designs. New Delhi, India: New Age International Publisher. Singh, G. N. (003). On the improvement of product method of estimation in sample surves. Journal of the Indian Societ of Agricultural Statistics, 5(3): 7-5. Singh, H. P., & Tailor, R. (003). Use of known correlation coefficient in estimating the finite population means. Statistics in Transition, (4):

24 SUBRAMANI & PRABAVATH Singh, H. P., & Tailor, R. (005). Estimation of finite population mean using known correlation coefficient between auxiliar characters. Statistica, 5: Singh, H. P., Tailor, R., & Kakran, M. S. (004). Improved estimators of population mean using power transformation. Journal of the Indian Societ of Agricultural Statistics, 58(): Singh, M. P. (95). On the estimation of ratio and product of the population parameters. Sankhā B, 7: Singh, M. P. (97a). Multivariate product method of estimation for finite populations. Journal of the Indian Societ of Agricultural Statistics, 3, Singh, M. P. (97b). Ratio cum product method of estimation. Metrika, : Sisodia, B. V. S., & Dwivedi, V. K. (98). A modified ratio estimator using coefficient of variation of auxiliar variable. Journal of the Indian Societ of Agricultural Statistics, 33():3-8. Srivenkataramana, T. (980). A dual to ratio estimator in sample surves. Biometrika, 7: Srivenkataramana, T., & Trac, D. S. (98). An alternative to ratio method in sample surves. Annals of the Institute of Statistical Mathematics, 3: -0. Subramani, J. (03).Generalized Modified Ratio Estimator for Estimation of Finite Population Mean. Journal of Modern Applied Statistical Methods, (), -55. Subramani, J., & Kumarapandian, G. (0a). Estimation of population mean using known median and co-efficient of skewness. American Journal of Mathematics and Statistics, (5): Subramani, J., & Kumarapandian, G. (0b). Estimation of population mean using co-efficient of variation and median of an auxiliar variable. International Journal of Probabilit and Statistics, (4): -8. Subramani, J., & Kumarapandian, G. (0c). Modified ratio estimators using known median and co-efficient of kurtosis, American Journal of Mathematics and Statistics, (4): Subramani, J., & Kumarapandian, G. (03). A new modified ratio estimator for estimation of population mean when median of the auxiliar variable is known. Pakistan Journal of Statistics and Operation Research, 9(),

25 TWO PARAMETER MODIFIED RATIO ESTIMATORS Tailor, R., Parmar, R., Kim, J. M., & Tailor, R. (0). Ratio-cum-Product estimators of population mean using known population parameters of auxiliar variable. Communication of the Korean Statistical Societ, 8(): Trac, D. S., Singh, H. P., & Singh, R. (99). An alternative to the ratiocum- product estimator in sample surves. Journal of Statistical Planning and Inference, 53: Upadhaa, L. N., & Singh, H. P. (999). Use of transformed auxiliar variable in estimating the finite population mean. Biometrical Journal, 4(5): 7-3. an, Z., & Tian, B. (00). Ratio method to the mean estimation using coefficient of skewness of auxiliar variable. ICICA 00, Part II, CCIS 0: 03-0.