E cient ratio-type estimators of nite population mean based on correlation coe cient

Transcription

1 Scientia Iranica E (08) 5(4), 36{37 Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering E cient ratio-type estimators of nite population mean based on correlation coe cient M. Irfana,b, M. Javeda,b, and Zh. Lina a. Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou 3007, China. b. Department of Statistics, Government College University, Faisalabad, Pakistan. Received 8 September 06 received in revised form 4 February 07 accepted 0 May 07 KEYWORDS Abstract. We proposed e cient families of ratio-type estimators to estimate nite. Introduction The ratio estimator suggested by Cochran [] plays an important role in the scenario of positive (high) correlation among study and auxiliary variables. One of the hottest issues in the theory of sample survey is the estimation of nite population mean with di erent sampling techniques. Extensive work has been done on this issue using auxiliary information, such as population median, mean, quartiles, deciles, coe cient of variation, coe cient of correlation, coe cient of skewness, coe cient of kurtosis, etc. Various estimators or classes of estimators have been proposed by several authors, including Sisodia and Dwivedi [], Upadhyaya and Singh [3], Singh and Tailor [4], Kadilar and Cingi [5,6], Singh et al. [7,8], Gupta and Shabbir [9], Koyuncu and Kadilar [0,], Yan and Tian [], Haq and Shabbir [3], Singh and Solanki [4], Yadav and Kadilar [5], Subramani and Kumarapandiyan [6, 7], Subramani and Prabavathy [8], Yadav et al. [9], Khan et al. [0], Kumar [], Irfan et al. [], Walia et al. [3] etc., as solutions to this issue to get improved results. Let \N " be the population size and \n" be the sample size such that n < N is drawn under simple random sampling without replacement (SRSWOR) scheme. Assume y and x are the study and auxiliary variables, respectively. Here, yi and xi represent the values of y and x for the ith unit of the population. Some important formulae used in this manuscript are given as follows: *. Corresponding author. addresses: mirfan@zju.edu.cn (M. Irfan) mariastat@zju.edu.cn (M. Javed) zlin@zju.edu.cn (Zh. Lin) Population mean of study variable: Auxiliary variable Bias Correlation coe cient E ciency Mean squared error Ratio-type estimators. doi: 0.400/sci population mean using known correlation coe cient between study variable and auxiliary variable by adopting Singh and Tailor's [Singh, H.P., and Tailor, R. \Use of known correlation coe cient in estimating the nite population means", Statistics in Transition, 6(4), pp (003)] estimator and Kadilar and Cingi's [Kadilar, C., and Cingi, H. \An improvement in estimating the population mean by using the correlation coe cient", Hacettepe Journal of Mathematics and Statistics, 35() pp (006a)] class of estimators in simple random sampling without replacement. The newly proposed estimators behaved e ciently as compared to the common unbiased estimator, traditional ratio estimator, and the other competing estimators. Bias, mean squared error, and minimum mean squared error of the proposed ratio-type estimators were derived. Moreover, theoretical ndings were proven with cooperation of two real data sets. 08 Sharif University of Technology. All rights reserved. N Y = N i= yi :

2 36 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{37 Population mean of auxiliary variable: = N N i= x i : Sample mean of study variable: y = n n i= y i : Sample mean of auxiliary variable: x = n n i= x i : Sampling fraction: f = n N : Ratio of population mean of study variable to population mean of auxiliary variable: R = Y : Constant term: = : n N Relative error terms and their expectations have been dened in order to get the bias, Mean Squared Error (MSE), and minimum MSE for the existing and proposed ratio-type estimators in this way. Let 0 = y Y Y and = x such that E( i) = 0 for i = 0 and, where E(:) represents the mathematical expectation E( 0) = S y Y E( ) = S x S y = (N ) N i=(y i Y ) S x = (N ) N i=(x i ) E( 0 ) = S yx Y some well-known traditional and existing estimators in SRSWOR are provided. Commonly used unbiased estimator of Y is: ^Y = y: () Estimator in Eq. () has the following mean squared error or variance: MSE( ^ Y ) = V ( ^ Y ) = S y : () The traditional ratio estimator of population mean suggested by Cochran [] is dened as: ^Y R = y x 6= 0: (3) x Bias and MSE of ^ YR up to the rst degree of approximation are, respectively, given as: Bias ( ^ YR ) Y C x yx C y C x (4) MSE ( ^ YR ) Y C y + C x( ') (5) ' = yx C y C x C y = S y Y C x = S x yx = (S y S x ) S yx : Given below is the introduction of some other competing ratio-type estimators that take known correlation coecient between study and auxiliary variables into account... Singh and tailor estimator In order to estimate nite population mean, Singh and Tailor [4] proposed ratio estimator by utilizing the known correlation coecient yx : + yx ^Y ST = y : (6) x + yx Bias and MSE up to the rst order of approximation of ^ YST in Eq. (6) are given by: Bias ( ^ YST ) Y C x yx C y C x (7) S yx = (N ) N i=(y i Y )(xi ): The motivation behind this manuscript is to develop new ecient families of ratio-type estimators to estimate population mean using SRSWOR scheme with the help of known correlation coecient between study variable and auxiliary variable. In the following, MSE( ^ YST ) Y C y + C x( ') (8) = + yx :

3 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{ Kadilar and Cingi estimators Kadilar and Cingi [4] proposed a class of ratio estimators to estimate population mean under SRSWOR in the light of the work done by Upadhyaya and Singh [3] and Singh and Tailor [4]. Their proposed class of estimators is: Cx + yx ^Y KC = y (9) xc x + yx yx + C x ^Y KC = y (0) x yx + C x (x) + yx ^Y KC3 = y () x (x) + yx ^Y KC4 = y yx + (x) x yx + (x) () where yx, C x, and (x) are the correlation coecient between study and auxiliary variables, coecient of variation, and coecient of kurtosis of the auxiliary variable, respectively. Bias and MSE of the above-mentioned estimators in Eqs. (9) to () are listed below: Bias ( ^ YKCi ) Y C x i ( i ') i = 3 4: (3) MSE ( ^ YKCi ) Y C y + C x i ( i ') i = 3 4 (4) = 3 = C x C x + yx = (x) (x) + yx 4 = yx yx + C x yx yx + (x) : Kadilar and Cingi [5] dened another class of ratiotype estimators as: ^Y KCi = y + b( x) i i = (5) = + yx x + yx 3 = yx + C x x yx + C x 5 = yx + (x) x yx + (x) : = C x + yx xc x + yx 4 = (x) + yx x (x) + yx The following are the expressions for bias and MSE of the ^ Y KCi estimators: Bias ( ^ Y KCi ) Y C x i i = (6) MSE ( ^ Y KCi ) Y h i C x + C y( = 3 = 5 = i yx) i = (7) + yx = yx yx + C x 4 = yx yx + (x) : C x C x + yx (x) (x) + yx. Proposed families of ratio-type estimators Inspired by Singh and Tailor [4] and Kadilar and Cingi [4,5], we propose ecient families of ratiotype estimators, i.e., ( ^ YP i, i.e., i = 3 5 ::: ) and ( ^ YP j, i.e., j = 4 6 ::: 6), to estimate population mean using the correlation coecient between study and auxiliary variables under SRSWOR scheme... The rst proposed family of ratio-type estimators Cx + yx ^Y P = ky xc x + yx (x) + yx ^Y P 3 = ky x (x) + yx (x) + yx C x ^Y P 5 = ky x (x) + yx C x Cx Cx+yx (8) (x) (x)+yx (9) (x) (x)+yxcx (0) yx yx(x) (x) + C x yx(x)+cx ^Y P 7 = ky () x yx (x) + C x + yx ^Y P 9 = ky x + yx Cx (x) + yx ^Y P = ky xc x (x) + yx +yx () Cx(x) Cx(x)+yx (3) where k is the selected appropriate constant whose value is determined later in Theorem. and population mean, C x population coecient of variation, yx population correlation coecient, and (x) population coecient of kurtosis are the known parameters.

4 364 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{37 Theorem. The properties of ^ YP i, i.e., i = 3 5 ::: are: ) Bias ( ^ YP i ) Y (k ) + Y k ( 4 i + 3 i )C x i yx C y C x (4) ) MSE ( ^ YP i ) Y Ai Ai + E( A i A 0) i + ( 4 i + 3 i )E( ) 4 i E( 0 ) A i A i ( 4 i + 3 i )E( ) i E( 0 ) (5) 3) MSE min ( ^ YP i ) Y = 5 = 9 = C x C x + yx 3 = (x) (x) + yx C x 7 = + yx = A i (6) A i (x) (x) + yx yx (x) yx (x) + C x C x (x) C x (x) + yx : To prove Eqs. (4)-(6), the proposed family of ratiotype estimators ( ^ YP i, i.e., i = 3 5 ::: ) can be written in terms of 0 i s as: or: ^Y P i = k Y ( + 0 )[ + i ] i (7) ^Y P i = k Y ( + 0 ) i + 3 i ( i + ) : (8) Simplifying the R.H.S. of Eq. (8) up to the rst degree of approximation and, then, subtracting Y from both sides, we get: ^Y P i Y = Y k + k 0 k i + k( i 4 + i 3 ) k i 0 : (9) Bias and the MSE of the proposed family of ratio-type estimators ( ^ YP i, i.e., i = 3 5 ::: ) are, respectively, given as: Bias( ^ YP i ) =E( ^ YP i Y ) Y (k ) + Y k ( 4 i + 3 i )C x i yx C y C x (30) MSE( ^ YP i ) = ( ^ YP i Y ) Y (k ) + k E( 0) + ( 4 i + 3 i )E( ) 4 i E( 0 ) k ( i 4 + i 3 )E() i E( 0 ) : (3) To get the optimal value of k, dierentiating Eq. (3) with respect to k and equating to zero, we have: + ( i 4 k = + i 3)E( ) i E( 0 ) + [E(0 ) + ( i 4 + i 3)E( ) 4 i E( 0 )] + k = ( 4 i + i 3)C x i yxc y C x + Cy + ( i 4 + i 3)C x 4 i yxc y C x k = A i A i A i = + ( 4 i + 3 i )C x i yx C y C x A i = + C y + ( 4 i + 3 i )C x 4 i yx C y C x : Putting the optimal values of k in Relation (3), we have MSE as: + MSE( ^ YP i ) Y Ai A i + ( 4 i + 3 i )E( ) 4 i E( 0 ) Ai A i E( 0) A i A i ( 4 i + 3 i )E( ) i E( 0 ) : (3) Simplifying Relation (3), we have the minimum MSE of the proposed ratio-type estimators: MSE min ( ^ YP i ) Y A i A i : (33)

5 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{ Second proposed family of ratio-type estimators ^Y P = t 4 yx + R y x yx + R yx + RC x ^Y P 4 = t y x yx + RC x ^Y P 6 =t y ^Y P 8 =t y ^Y P 0 = t yx + R (x) x yx + R (x) yx yx+r + b( x) 3 5 (34) yx yx+rcx +b( yx yx+r(x) x) (35) + b( x) (36) yx + RC x (x) x yx + RC x (x) yx yx+rcx(x) + b( x) (37) 4 + Ryx 3 +Ryx y + b( x) 5 (38) x + R yx + Ryx C x ^Y P = t y x + R yx C x ^Y P 4 =t y ^Y P 6 =t y + Ryx (x) x + R yx (x) +RyxCx +b( +Ryx(x) x) (39) + b( x) (40) + Ryx C x (x) x + R yx C x (x) +RyxCx(x) + b( x) (4) where t is the unknown constant, which is to be determined later on, and, Cx, yx, and (x) are the known population parameters. Theorem. The properties of ^ YP j, i.e., j = 4 6 ::: 6, are: ) Bias( ^ YP j ) Y (t ) + Y t ('4 j + ' 3 j)c x ' j yx C y C x (4) ) MSE( ^ YP j ) A3j Y Y + A3j Y E( ) + + ' 3 j Y + ' jb Y E( ) ' 4 j Y + b 4' j Y + b Y E( 0 ) A3j ' 4 j Y + ' 3 jy E( ) ' jy E( 0 ) : (43) " # 3) MSE min ( ^ YP j ) Y A 3j (44) ' = ' 6 = ' 0 = ' 4 = yx yx + R ' 4 = yx yx + R (x) ' 8 = + R yx ' = + R yx (x) ' 6 = yx yx + RC x yx yx + RC x (x) + R yx C x + R yx C x (x) : To prove Relations (4) to (44), the proposed estimators ( ^ YP j, i.e., j = 4 6 ::: 6) may be written in terms of i 0 s as follows: or: ^Y P j = t ( Y + Y 0 )( + ' j ) ' j b ^Y P j = t b = yxs y S x : ( Y + Y 0 ) ' j + '3 j(' j + ) b (45) Subtracting Y from Eq. (45) and expanding it up to the rst degree of approximation, we obtain: ^Y P j Y = t Y + t Y 0 t Y ' j tb + t Y (' 4 j +' 3 j) t Y ' j 0 Y : (46)

6 366 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{37 The bias and the MSE of ^ YP j i.e., j = 4 6 ::: 6, up to the rst degree of approximation are given as: Bias ( ^ YP j ) Y (t MSE( ^ YP j ) ) + Y t ('4 j + ' 3 j)c x ' j yx C y C x (47) (t Y Y ) + t Y E( 0) + ' 4j Y + b + ' 3j Y + ' jb Y E() 4' j Y + b Y E(0 ) ' 4 t jy +' 3 jy E() ' jy E( 0 ) : (48) Dierentiating Eq. (48) with respect to t and equating to zero, we have t as shown in Box I, A 3j = Y + = Y + (' 4 j + ' 3 j)r S x ' jrs yx S y + ' 4 jr + b + ' 3 jr + ' jbr S x f4' jr + bgs yx : Inserting the value of t in Eq. (48), we obtain: MSE( ^ YP j ) A3j Y Y + A3j Y E( 0) + ' 4 j Y + b + ' 3 j Y + ' jb Y E( ) 4' j Y + b Y E(0 ) A 3j ' 4 A jy +' 3 jy E() ' jy E( 0 ) : 4j (49) Thus, the minimum MSE is as follows: " # MSE min ( ^ YP j ) Y A 3j : (50) Interesting note Many more ratio-type estimators based on correlation coecient can be formulated by taking dierent measures of i and ' j. 3. Eciency of the proposed estimators This section deals with the derivation of algebraic situations under which the proposed estimators will have minimum MSE as compared to unbiased estimator (sample mean), traditional ratio estimator, Singh and Tailor's [4] estimator, and Kadilar and Cingi's [4,5] classes of estimators. Theorem 3.. ^ YP i (i.e., i = 3 5 ::: ) perform better than ^ Y if: Cy + A i > : (5) A i. YP ^ j (i.e., j = 4 6 ::: 6) perform better than Y ^ if: " # Cy + A 3j Y > : (5). By comparing Relations () and (33): MSE min ( ^ YP i ) < MSE( Y ^ ) Y A i < S A y i t = t = Y + f' 4 jy + ' 3 jy ge() ' jy E( 0 ) 4' jy + b Y E(0 ) Y + Y E( 0 ) + ' 4 j Y + b + ' 3 j Y + ' j b Y E( ) Y + (' 4 j + ' 3 j )R Sx ' j yx RS S x 4' j R + b S yx Y + S y + ' 4 j R + b + ' 3 j R + ' j br t = A 3j Box I

7 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{ Cy + A i > : A i. By comparing Relations () and (50): MSE min ( ^ YP j ) < MSE( Y ^ ) " # Y A 3j < S A y 4j " C y + Theorem 3. # A 3j Y > :. ^ YP i (i.e., i = 3 5 ::: ) perform better than traditional ratio estimator, ^ YR, if: C y + C x( ') + A i A i > : (53). ^ YP j (i.e., j = 4 6 ::: 6) perform better than traditional ratio estimator, ^ YR, if: C y + C x( ') + A 3j Y > : (54). By comparing Relations (5) and (33): MSE min ( ^ YP i ) < MSE( ^ YR ) Y C y + C x( A i < Y Cy + C A x( ') i ') + A i A i > :. By comparing Relations (5) and (50): MSE min ( ^ YP j ) < MSE( ^ YR ) " Y # A 3j < Y Cy + C A x( ') 4j C y + C x( ') + A 3j Y > : Theorem 3.3. ^ YP i (i.e., i = 3 5 ::: ) perform better than Singh and Tailor's [4] estimator, ^ YST, if: C y + C x( ') + A i A i > : (55). ^ YP j (i.e., j = 4 6 ::: 6) perform better than Singh and Tailor's [4] estimator, ^ YST, if: C y + C x( ') + A 3j Y > : (56). By comparing Relations (8) and (33): MSE min ( ^ YP i ) < MSE( ^ YST ) Y C y + C x( A i < Y Cy + C A x( ') i ') + A i A i > :. By comparing Relations (8) and (50): MSE min ( ^ YP j ) < MSE( ^ YST ) " Y # A 3j < Y Cy + C A x( ') 4j C y + C x( ') + A 3j Y > : Theorem 3.4. ^ YP i (i.e., i = 3 5 ::: ) perform better than Kadilar and Cingi's [4] estimators, ^ YKCi, if: C y + C x i ( i ') + A i A i > : (57). ^ YP j (i.e., j = 4 6 ::: 6) perform better than Kadilar and Cingi's [4] estimators, ^ YKCi, if: C y + C x i ( i ') + A 3j Y > : (58). By comparing Relations (4) and (33): MSE min ( ^ YP i ) < MSE( ^ YKCi ) Y C y + C x i ( i A i < Y Cy + C A x i ( i ') i ') + A i A i > :. By comparing Relations (4) and (50): MSE min ( ^ YP j ) < MSE( ^ YKCi ) " Y # A 3j < Y Cy + C A x i ( i ') 4j C y + C x i ( i ') + A 3j Y > :

8 368 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{37 Theorem 3.5. ^ YP i (i.e., i = 3 5 ::: ) perform better than Kadilar and Cingi's [5] estimators, ^ Y KCi, if: h i C x + C y( yx) i + A i A i > : (59). YP ^ j (i.e., j = 4 6 ::: 6) perform better than Kadilar and Cingi's [5] estimators, Y ^ KCi, if: h i i Cx + Cy( yx) + A 3j Y > : (60). By comparing Relations (7) and (33): MSE min ( ^ YP i ) < MSE( ^ Y KCi ) Y A h i < Y i Cx + C A y( yx) i h i Cx + Cy( i yx) + A i > : A i. By comparing Relations (7) and (50): MSE min ( ^ YP j ) < MSE( Y ^ KCi ) " Y # A h i 3j < Y i Cx + C A y( yx) 4j h i Cx + Cy( 4. Empirical study i yx) + A 3j Y > : This section evaluates the performance of the proposed families of ratio-type estimators, i.e., ( ^ YP i i.e., i = 3 5 ::: ) and ( ^ YP j i.e., j = 4 6 ::: 6), with competitive estimators. For this purpose, we consider two natural populations described below. i Population (source: Kadilar and Cingi [4]): The data relates to 04 villages of East Anatolia Region of Turkey in 999. The following variables are taken into consideration: y The level of apple production (in 000 tones) x The number of apple trees. Values of dierent required parameters are: N = 04, n = 0, yx = 0:865, Y = 65:37, = 3:93, Cy = :866, C x = :653, and (x) = 7:5. Population (source: Murthy [6]): The variables are dened as follows: y The output for 80 factories x The number of workers. Values of dierent required parameters are: N = 80, n = 0, yx = 0:950, Y = 5:864, = :853, C y = 0:354, C x = 0:9484, S y = 8:3569, S x = :704, and (x) = :3005. Consider the MSE given in Section and derived in Section the values for existing and proposed families of estimators are computed and placed in Tables and. The following are some important ndings observed in Tables and : ˆ It is important to note that the traditional ratio estimator is more ecient than the suggested class of estimators by Kadilar and Cingi [5], i.e., ^ Y KC, ^Y KC, ^ Y KC3, ^ Y KC4, and ^ Y KC5, for both real populations discussed in this manuscript ˆ It can be concluded that proposed families of ratiotype estimators are more ecient as they have lesser values of MSE as compared to the usual unbiased estimator, traditional ratio estimator, Singh and Tailor's [4] estimator, and Kadilar and Cingi's [4,5] classes of estimators ˆ It is perceived in both populations that the two proposed estimators ^ YP and ^ YP 6 have the least Table. MSEs of the competing estimators for Population. Estimators MSE Estimators MSE Estimators MSE Estimators MSE ^Y Y ^ KC YP ^ YP ^ ^Y R ^ Y KC ^ YP ^ YP ^Y ST ^ Y KC ^ YP ^ YP ^Y KC ^ Y KC ^ YP ^ YP ^Y KC ^ Y KC ^ YP ^ YP ^Y KC ^ YP ^ YP ^Y KC ^ YP ^ YP Bold value indicates minimum MSE.

9 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{ Table. MSEs of the competing estimators for Population. Estimators MSE Estimators MSE Estimators MSE Estimators MSE ^Y.6366 Y ^ KC YP ^ YP ^ ^Y R ^ Y KC ^ YP ^ YP ^Y ST ^ Y KC ^ YP ^ YP.057 ^Y KC ^ Y KC ^ YP 9.64 ^ YP ^Y KC ^ Y KC ^ YP.0783 ^ YP ^Y KC3.570 YP ^ 6.7 YP ^ ^Y KC4.666 YP ^ YP ^ Bold value indicates minimum MSE. Table 3. PREs of the competing estimators for Population. Proposed Existing estimators estimators ^Y ^Y R ^Y ST ^Y KC ^Y KC ^Y KC3 ^Y KC4 ^Y KC ^Y KC ^Y KC3 ^Y KC4 ^Y KC5 ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P MSE values ( and.0645) among all the proposed estimators ˆ It is interesting to note that the Kadilar and Cingi's [4] estimators, ^ YKC and ^ YKC3, are the special cases of the proposed estimators, ^ YP and ^Y P 3. When we put the values of k = and i =, where i = and, in ^ YP and ^ YP 3, these estimators give the same result as the estimators ^ YKC and ^Y KC3. This study calculated the Percentage Relative Eciencies (PREs) for both populations under consideration to prove the dominance of the proposed families of ratio-type estimators, i.e., ( ^ YP i, i.e., i = 3 5 ::: ) and ( ^ YP j, i.e., j = 4 6 ::: 6), over the existing estimators, i.e., ^ Y, ^ YR, ^ YST, ^ YKC, ^ YKC, ^Y KC3, ^ YKC4, ^ Y KC, ^ Y KC, ^ Y KC3, ^ Y KC4, and ^ Y KC5. Here, the Percentage Relative Eciency (PRE) is the ratio of MSE of the existing estimators (e) to the MSE of the proposed ratio-type estimators (p) and is given by: PRE(e p) = MSE(e) 00: (6) MSE(p) Tables 3 and 4 provide the PREs calculated for both populations. It is revealed in Tables 3 and 4 that the proposed families of ratio-type estimators are more competent than the existing estimators for both populations as they have higher values. To get deeper insight, another important tool, i.e., Relative Root Mean Square Error (RRMSE), is computed in this study. It is the most useful measure to compare the precision of the estimators (see, e.g., [,,7,8]). The RRMSE of an estimator can be calculated by the relation given below: RRMSE = q where MSE may be dened as: MSE( ^) = n MSE( ^) (6) n i= ( ^ ) (63) where ^ is the estimate of on the ith sample. It is quite obvious from the values of RRMSE shown in Tables 5 and 6 that the proposed families of ratio-type estimators perform better than the existing ones discussed in this article.

10 370 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{37 Table 4. PREs of the competing estimators for Population. Proposed Existing estimators estimators ^Y ^Y R ^Y ST ^Y KC ^Y KC ^Y KC3 ^Y KC4 ^Y KC ^Y KC ^Y KC3 ^Y KC4 ^Y KC5 ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P ^Y P Table 5. RRMSEs of the competing estimators for Population. Estimators RRMSE Estimators RRMSE Estimators RRMSE Estimators RRMSE ^Y Y ^ KC YP ^ YP ^ ^Y R Y ^ KC YP ^ YP ^ ^Y ST Y ^ KC YP ^ YP ^ ^Y KC 0.88 Y ^ KC YP ^ YP ^ ^Y KC Y ^ KC YP ^ 0.86 YP ^ ^Y KC YP ^ YP ^ ^Y KC YP ^ YP ^ Table 6. RRMSEs of the competing estimators for Population. Estimators RRMSE Estimators RRMSE Estimators RRMSE Estimators RRMSE ^Y Y ^ KC 0.48 YP ^ YP ^ ^Y R 0.40 Y ^ KC YP ^ YP ^ ^Y ST 0.08 Y ^ KC YP ^ YP ^ ^Y KC Y ^ KC YP ^ YP ^ ^Y KC Y ^ KC YP ^ YP ^ ^Y KC YP ^ YP ^ ^Y KC YP ^ YP ^ Conclusion In this article, we proposed improved families of ratiotype estimators of population mean under simple random sampling without replacement (SRSWOR) scheme using correlation coecient between study and auxiliary variables. We obtained bias, Mean Square Error (MSE), and minimum MSE formulae of the proposed families of ratio-type estimators and compared them theoretically with those of the traditional and exiting modied ratio estimators in the literature. It was found that the newly proposed estimators were more ecient than the traditional estimators, such as usual unbiased and ratio, i.e., Y ^ and YR ^, and existing modied ratio estimators, i.e., ^ YST (Singh and Tailor [4]), YKCi ^ (Kadilar and Cingi [4]), and Y ^ KCi (Kadilar and Cingi [5]), in terms of Mean Squared Error (MSE), Percentage Relative Eciency (PRE), and Relative Root Mean Square Error (RRMSE). It was also empirically observed that the proposed families of ratio-type estimators performed better than the traditional and existing modied ratio estimators by using two natural population data sets. Hence, we strongly suggest the use of our newly proposed ratiotype estimators over the existing ratio-type estimators used in this study for future work. Acknowledgments The authors are heartily thankful to the unknown learned referees for their valuable remarks and encouragements.

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12 37 M. Irfan et al./scientia Iranica, Transactions E: Industrial Engineering 5 (08) 36{37 Biographies Muhammad Irfan is a doctoral student of Statistics at Zhejiang University, China. He has been working as lecturer in the Department of Statistics, Government College University, Faisalabad, Pakistan, since 004. He has research papers published in reputed journals. His areas of interest are survey sampling, probability distributions, and time series analysis. Maria Javed is a doctoral student of Statistics at Zhejiang University, China. She has been working as lecturer in the Department of Statistics, Government College University, Faisalabad, Pakistan, since 004. She has more than 8 research publications in international research journals. Her elds of work are probability distributions and survey sampling. Zhengyan Lin is a Professor of Statistics at Zhejiang University, China. He has over 30 refereed publications and 8 edited book volumes, and has guided research students at all levels of the curriculum, including undergraduate and PhD students. He has received external funding from National Natural Science.