Comparison of Two Ratio Estimators Using Auxiliary Information

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1 IOR Journal of Mathematics (IOR-JM) e-in: , p-in: Volume, Issue 4 Ver. I (Jul. - Aug.06), PP omparison of Two Ratio Estimators Using Auxiliar Information Bawa, Ibrahim, Makada, Audu Federal Inland Revenue ervices, Abuja. tatistics Department, Niger tate Poltechnic Zungeru. Abstract: This stud is conducted to compare two ratio estimators that use auxiliar information. The estimators are Olkin average ratio estimator and Kadilar-ingi estimator. Efficiencies of the estimators are investigated theoreticall and using data from Federal Inland Revenue ervice Revenue House tore. The result shows Kadilar ingi estimator is more efficient than Olkin estimator. Kewords: Auxiliar variable,ratio Estimators,Efficienc, omparison. I. Indroduction The use of auxiliar variable in surve sampling has old histor. The auxiliar variable has been efficientl used b numerous surve statisticians to increase the precision of estimates. Laplace (80) has used the auxiliar information in estimating the population of France. The use of auxiliar information in allocating probabilities of selection has been demonstrated b Hansen and Hurwitz (943). The auxiliar information also plas ver significant role in number of estimating techniques in surve sampling. ochran (94) demonstrated the use of auxiliar information in defining the ratio estimator of population mean and total. ochran, (94) also introduced the regression estimator as more efficient method of estimation in surve sampling. rivastava (967) had shown the efficient use of auxiliar information in improving the performance of estimator. The use of multiple auxiliar variables in estimation has been introduced b Raj (965). everal ratio tpe and chain ratio tpe estimators have been discussed in the literature using the information on single and multiple auxiliar variables. The estimation of the population mean is a persistent issue in sampling practice and man efforts have been made to improve the precision of the estimates. The literature on surve sampling describes a great variet of techniques for using auxiliar information b means of ratio, product and regression methods. Particularl, in the presence of multi-auxiliar variables, a wide variet of estimators have been discussed, following different ideas, and linking together ratio, product or regression estimators, each one exploiting the variables one at a time. Olkin (958) was the first author to deal with the problem of estimating the mean of a surve variable when auxiliar variables are made available. He suggested the use of information on more than one supplementar characteristic, positivel correlated with the stud variable, considering a linear combination of ratio estimators based on each auxiliar variable separatel. The coefficients of the linear combination were determined so as to minimize the variance of the estimator. Analogousl to Olkin, ingh (967) gave a multivariate expression of Murth (964) product estimator, while Raj (965) suggested a method for using multi-auxiliar variables through a linear combination of single difference estimators. Moreover, ingh (967) considered the extension of the ratio-cum-product estimators to multi-supplementar variables, while Rao and Mudholkar (967) proposed a multivariate estimator based on a weighted sum of single ratio and product estimators. John (969) suggested two alternative multivariate generalizations of ratio and product estimators which actuall reduce to the Olkin s (958) and ingh (967) estimators. rivastava (97) considered a general ratio-tpe estimator which generates a large class of estimators including most of the estimators up to date. Man other contributions are present in sampling literature and, recentl, some new estimators appeared. Robinson (994), suggest a regression estimator ignoring some of the assumptions usuall adopted in the literature, rivastava, (97). Trac et al. (996) and Perri (004), when two auxiliar variables are available, suggested an alternative to ingh (965, 967) ratio-cum-product estimators. eccon and Diana (996) gave a multivariate extension of the Naik and Gupta (99) univariate class of estimators. Agarwal et al. (997), moving from Raj (965), illustrated a new approach to form a multivariate difference estimator which does not require the knowledge of an population parameters. Abu-Daeh et al. (003) introduced two estimators which are indeed members of the class proposed b rivastava (97), while Kadilar and ingi (004, 005) analzed combinations of regression tpe estimators in the case of two auxiliar variables. In the same situation, Perri (005) proposed some new estimators obtained from ingh (965, 967) ones. DOI: / Page

2 omparison of Two Ratio Estimators Using Auxiliar Information Tpicall, almost all the estimators are based on the known means of auxiliar variables are nonlinear in the auxiliar means and the efficienc is studied through the first order approximation of their mean square error Wolter (985). But, in spite of the wide variet of estimators, man estimators exhibit the same performance at the first order of approximation. Moreover, the generall do not consider other sources of supplementar information such as that one provided b the moments of order greater than the first. As a matter of fact, some works deal with the problem of estimating the unknown mean of a stud variable b using the means and variances of the auxiliar variables, but onl in a two-phase simple random sampling scheme, Diana and Tommasi (003), for a review. The use of multi-auxiliar information about means and variances is scarcel considered outside the double sampling Das and Tripathi (98) in the case of a single auxiliar variable. In addition most of the authors either propose a new estimator or developed the existing estimator leaving the concept of comparison among the existing estimators silent. In this regard an attempt is made to compare two existing estimators to determine the most efficient. II. Materials And Methods. source Of Data The data used in this stud were collected from Federal Inland Revenue ervice (FIR) Abuja. The data consist of 00 observations of overhead expenditure on office routine electronic equipment s and 00 observations of overhead expenditure on office routine furniture equipment s for the ear 0. The data is based on the cost of items (Y), manufacture length ( ) and das in the store ( ). Large sample size n > 30 units will be selected.. Olkin (958) onventional Multivariate Ratio Estimator Olkin (958) presented the multivariate average ratio estimator using information of two auxiliar variables x and x to estimate the population mean (Y ), as follows: Y w w x x (..) Where denotes the sample mean of the variable of interest ; sample and the population means of the variable w w. The Olkin estimator becomes; ( Y ) (..) x x xi and x i (i =, ) denote respectivel the x i ; and w, w are the weights that satisf the condition: The stud shows the propert of as an estimator ofy. Thus from.. taking expectation; E( Y ) E( w w ) x x ( ) ( ) EY ( ) we we x x ince E( x), then E( Y ) w ( ) w ( ) E( Y ) w w E( Y ) ( w w) But w w E( Y ). This is unbiased The ME of this estimator is given b ingh (986) as: DOI: / Page

3 omparison of Two Ratio Estimators Using Auxiliar Information f ME( Y ) Y { w w w w w n w x } (..3) x x x x x xx x x w w (..3) becomes; f ME( Y ) Y {4 4x 4 } x x x x xx x x x 4n (..4) n f,, x and x N, when where n and N are the sample and the population sizes respectivel; coefficient of variation of Y, and between Y and, Y and, and respectivel and x x, xx respectivel, Abu-Daeh et al. (003). denote the denote the correlation coefficient.3 The Kadilar And ingi (004) Estimator The Kadilar and ingi Regression estimator, stud used the following estimator: ( x ) ( x ) YREG ( pr) (.3.) x x Note β i 0, ( i =, ). Where and are the weights that satisf the condition: coefficients for x and x respectivel. ; The stud shows the propert of as an estimator ofy. Thus from 3.4. taking expectation; ( x ) ( x ) E( Y ) E( ) x x and are the regression E( Y ) E( { x }) E( { x }) x x E( Y ) E( ) E( ) E( x ) E( ) E( ) E( x ) REG ( pr ) x x x x x x E( Y ) E( Y ) E( Y ) ( ) E( Y ) But REG( pr ). This is also unbiased. The ME of this estimator can be found using Talor series method defined b T ME( Y ) d d (.3.) where REG( pr ) (, x, x ) (, x, x ) (, x, x ) d [ ] Y,, Y,, Y,, x x DOI: / Page

4 omparison of Two Ratio Estimators Using Auxiliar Information x x f x x x x n x x x x Here, x and x denote the covariance between Yand,Yand, and respectivel and x x, x x, x x xx denote the covariance between Yand,Yand, and respectivel. ( Y ) In order to determine the minimum attainable mean square error, we take the partial derivatives of REG( pr ) with respect to components, x and x respectivel. (.3.) becomes; Y { x } { x } (.3.3) x x (, x, x ) Y, x, x x x = (, x, x ) x x x Y, x, x Y ( ) (, x, x ) x x x Y, x, x Y ( ) According to this definition, the stud obtained dfor the estimator as [ ( Y Y d ) ( ) ] where x x x x x and x x x x x The stud obtains ME of the estimator using (.3.) as follows: f Y Y Y ME( Y ( )) { ( ) ( ) ( ) n REG pr x x x Y Y Y ( ) ( )( ) (.3.4) x x x We can also write (.3.4) b f ME( YREG ( pr )) Y { ( x ) x x ( x ) x x n ( ) ( ) ( x x x x xx x xx x )} (.3.5) Where x x x x x x x x x x, Y x x and Y xx xx DOI: / Page

5 omparison of Two Ratio Estimators Using Auxiliar Information.4 omparison Of The Estimators This stud compares the mean square error (ME) of the Kadilar and ingi (004) estimator given in (.3.5) with the Olkin (958) estimator given in (3..4). The stud has the conditions as follows: ME( YREG ( pr) ) Efficienc ( E) ME( Y ) If: (i) ME( Y ) = ME( Y ), then Efficienc (E) =. Therefore, the two estimators are the same. (ii) ME( Y ) ME( Y ), then ME( Y ) is smaller than ME( Y ) ME( Y ) is more efficient than ME( Y ).. Therefore, (iii) Otherwise, ME( Y ) is more efficient than ME( Y ). Now the Kadilar and ingi (004) estimator will be more efficient than the Olkin (958) estimator if: ME( Y ) ME( Y ) 0 Now using (3..4) and (3.3.5) we have: ME( Y ) ME( Y ) 3 ( ) ( ) 4 4 x x x x x x ( ) ( ) xx x x x x x x ( ) ( ) ( x x x x xx x x x x x ) xx x x x xx (.3.6) < 0 When this condition is satisfied, the Kadilar and ingi (004) estimator will be more efficient than the Olkin (958) estimator. III. Results And Discussion 3. Introduction In this chapter empirical stud is conducted to see the performance of the Kadilar and ingi (004) estimator over the estimator proposed b Olkin (958). The values of variance-covariance are used to calculate efficienc of the estimator for various values of and. The data used for population is based on record on overhead expenditure of office routine electronic equipment s cost of items () and period of items from the date manufactured ( ) and period of items staed in the store ( ) for the ear 0. Then the samples of size 0 were selected. The data used for population is based on record on overhead expenditure of office routine furniture equipment s cost of items () and period of items from the date manufactured ( ) and period of items staed in the store ( ) for the ear 0. Then the samples of size 50 are drawn using simple random sampling method. DOI: / Page

6 omparison of Two Ratio Estimators Using Auxiliar Information Table 3.: ummar of Estimatesfor Populations and Table 3.: omputed ME for the Two Estimators However, And ME( YREG ( pr )) ME( Y ) ME( YREG ( pr )) ME( Y ).79 0 Table 3. shows efficienc comparison of the Kadilar and ingi (003) estimator with the Olkin (958) estimator. The entries of table 3. clearl indicate that the Kadilar and ingi (004) estimator has smaller ME than the Olkin (958) estimator in the both populations. These results are not surprising because the value of the condition (.3.6) are found as < 0 and ME( Y ). ME( Y ) ME( Y ) therefore, ME( Y ) is smaller than IV. onclusion The stud has found that Kandilar and ingi (004) multivariate ratio estimator has smaller mean square error than the Olkin (958) average ratio estimator has. Therefore, the stud concludes that the Kandilar and ingi (004) multivariate ratio estimator is more efficient than the Olkin (958) average ratio estimator for this population. References []. Abu-Daeh, W. A., Ahmed, M.., Ahmed, R. A., and Muttlak, H. A. (003) ome Estimators of finite population mean using auxiliar information, Applied Mathematics and omputation, 39, []. Adewara, A. A., Job, O., Abidoe, A. O. and Oeemi' G' M'. (008), Generalised Lnch Multivariate Estimate with K-parameters of Order n.journal of Mathematics ciences Vol. 9 No [3]. Agarwal,. K., harma, U. K., and Kashap,. (997) a new approach to use Multivariate auxiliar information in sample surves, Journal of tatistical Planning and Inference, 60, [4]. Bahl,. and Tuteja, R.K. (99): Ratio and Product tpe exponential estimator, Information and Optimization sciences, Vol.II, I, [5]. eccon,. and Diana, G. (996) lassi di stimatori di tipo rapporto in presenza di pi u variabiliausiliarie, Proceedings of Italian tatistical ociet, VIII cientifi Meeting, Rimini, 9-3 April 996, 9 6. [6]. ochran, W. G. (94): ampling theor when the sampling-units are of unequal sizes, Journal of American tatistics Association. 37, 99. [7]. ochran, W. G. (977): ampling Techniques, New York, John Wile and ons, 3 rd Edition. (IBN QA , ). [8]. Kadilar,. and ingi, H. (004) Estimator of a population mean using two auxiliar variables in simple random sampling, International Mathematical Journal, 5, [9]. Kadilar,. and ingi, H. (005) A new estimator using two auxiliar variables, Applied Mathematics and omputation, 6, [0]. Olkin, I. (958) Multivariate ratio estimation for finite populations, Biometrika, 45, DOI: / Page