College of Science, Inner Mongolia University of Technology, Hohhot, Inner Mongolia, China

Hindawi Mathematical Problems in Engineering Volume 07 Article ID 8704734 7 pages https://doiorg/055/07/8704734 Research Article Efficient Estimator of a Finite Population Mean Using Two Auxiliary Variables and Numerical Application in Agricultural Biomedical and Power Engineering Jingli Lu College of Science Inner Mongolia University of Technology Hohhot Inner Mongolia China Correspondence should be addressed to Jingli Lu; lujingli004@63com Received 3 January 07; Revised July 07; Accepted 5 July 07; Published 3 August 07 Academic Editor: Guido Ala Copyright 07 Jingli Lu This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited To improve the efficiency of an estimator with two auxiliary variables we propose a new estimator of a finite population mean under simple random sampling The bias and mean square error expressions of the proposed estimator have been obtained In a comparison study we found that the new estimator was consistently better than those of Abu-Dayyeh et al Kadilar and Cingi and Malik and Singh as well as the regression estimator using two auxiliary variables and that the minimum values of the previous three above reported estimators were equal We used four numerical examples in agricultural biomedical and power engineering to support these theoretical results thus enriching the theory of survey samples by the development of new estimators with two auxiliary variables Introduction In sampling theory it is a well-established phenomenon that supplementary information provided by auxiliary variables or auxiliary attributes often improves the accuracy of estimators of unknown population parameters Ratio- product- and regression-type estimators are three such methods For this reason some authors have exploited the use of auxiliary variables and attributes at the estimation stage to increase estimator efficiency For example the planting area and the proportion of good seeds in agricultural engineering are two important auxiliary variables when estimating average cotton output Similarly the breed of cow in animal husbandry engineering is an important auxiliary attribute when estimating average milk yield Thus auxiliary information can be used in the field of education biostatistics the medical research agricultural and biomedical engineering and so on In the literature some authors have proposed many efficient ratio- product- and regression-type estimators using one auxiliary variable or attribute including Singh and Vishwakarma [] Grover and Kaur [ 3] Singh et al [4] Singh and Solanki [5] and Gupta and Shabbir [6] More recently several authors have proposed efficient estimators of finite population mean using two variables or attributes including Abu-Dayyeh et al [7] Kadilar and Cingi [8] Malik and Singh [9] Sharma and Singh [0] and Muneer et al [] Although these studies are detailed and elaborated the formulas of minimum are not given and the difference of minimum values between these studies seems not to have been noticed In this paper we compare the estimators reported by Abu-Dayyeh et al [7] Kadilar and Cingi [8] and Malik and Singh [9] and introduce a new estimator with two auxiliary variables to estimate a finite population mean for the variable of interest We obtained bias and mean square error ( equations for the proposed estimator and we compared the new estimator against those with relatively high efficiencies An empirical study using four datasets in agricultural biomedical and power engineering was conducted and we obtained satisfactory results both theoretically and numerically The analysis of these issues is of great significance for understanding agricultural biomedical and power engineering Therefore the proposed estimator could be applied across a broad spectrum of sampling survey

Mathematical Problems in Engineering Materials and Methods Abu-Dayyeh Estimator Abu-Dayyeh et al [7] proposed thefollowingestimatorofpopulationmeanwhenthepopulation means X and X of the auxiliary variables were known: y AD y( X α ( X α ( x x where y denotes the sample means of the variable y x i and X i (i denote respectively the sample and the population means of the variable x i (i and α and α are real numbers The of y AD (y AD f n Y (C y +α C x +α C x α C y ρ yx α C y ρ yx +α α ρ x x where f n/n; n and N are respectively the number of units in the sample and the population; C y C x andc x are the coefficients of variation of Y X andx respectively;and ρ x x ρ yx andρ yx are the correlation coefficients between X and X Y and X andyand X respectively To minimize (y AD theoptimumvaluesofα and α are given by α C y (ρ yx ρ x x ρ yx ( ρx x α C y (ρ yx ρ x x ρ yx ( ρx x The minimum of y AD canbeshownas ( (3 The of ( f n Y (C y +α C x +α C x α C y ρ x x ρ yx α C y ρ x x ρ yx +α α ρ x x C y ρ yx C y ρ yx +C y ρ x x ρ yx ρ yx To minimize ( theoptimumvaluesofα and α are given by α C yρ x x (ρ yx ρ x x ρ yx ( ρx x α C yρ x x (ρ yx ρ x x ρ yx ( ρx x The minimum of can be shown as (6 (7 min ( δy C y L A (8 3 Malik and Singh Estimator Malik and Singh [9] proposed an estimator to estimate the population mean Y as follows: y exp ( X β x ( X β x X + x X + x (9 +b (X x +b (X x where β and β are real numbers The of ( δ[y (C y + 4 β C x + 4 β C x min (y AD δy C y L A (4 where δ ( f/n; L ρ x x ρ yx +ρ x x ρ yx ρ yx ρ yx ; A ρ x x Kadilar and Cingi Estimator Kadilar and Cingi [8] proposed an estimator using two auxiliary variables x and x toestimatethepopulationmeany as follows: + β β ρ x x β C y ρ yx β C y ρ yx +Y(β b C x X +β b C x X +β b ρ x x X +β b ρ x x X b C y ρ yx X b C y ρ yx X +b C x X +b b ρ x x X X +b C x X ] (0 y( X α ( X α +b x x (X x +b (X x where b S yx /S x and b S yx /S x ; S x and S x are the variances of Y X andx respectively;ands yx and S yx are the covariance between Yand X and Y and X respectively (5 To minimize ( theoptimumvaluesofβ and β are given by β C yρ x x (ρ yx ρ x x ρ yx ( ρx x β C yρ x x (ρ yx ρ x x ρ yx ( ρx x (

Mathematical Problems in Engineering 3 The minimum of canbeshownas min ( δy C y L A ( 4 The Regression Estimator Rao[]proposedanestimator using one auxiliary variable x toestimatethepopulation mean Y as follows: y Rao w y+w (X x (3 Similarly following Rao a regression estimator of Y using two auxiliary variables x and x isgivenby w y+w (X x +w 3 (X x (4 where w w andw 3 are real constants The of ( Y +w ( + δc y +δ{x C x w 3 + X w (X w +X w 3 ρ x x } Yw {Y +δc y (X w ρ yx + X w 3 ρ yx } (5 The optimum values of w w andw 3 obtained by minimizing (5 respectively are given by w A A+δC y L w YC y (ρ yx ρ x x ρ yx X (A + δc y L w 3 YC y (ρ yx ρ x x ρ yx X (A + δc y L The minimum of canbeshownas (6 min ( δy C y L A+δC y L (7 5 The Proposed Estimator Singh and Espejo [3] proposed an estimator using one auxiliary variable x to estimate the populationmeany as follows: Y SE y (X x + x (8 X Inspired by this work we propose a new estimator with two auxiliary variables as follows: k y+k (X x +k 3 (X x 4 + x X ( X x + x X where k k andk 3 are real constants ( X x (9 Let ε 0 y/y ε x /X andε x /X Under simple random sampling without replacement (SRSWOR we have the following expectations: E(ε 0 E(ε E(ε 0 E(ε 0 δc y E(ε δc x E(ε δc x E(ε 0 ε δρ yx C y E(ε 0 ε δρ yx C y E(ε ε δρ x x The proposed estimator can be rewritten as k Y(ε 0 + k X ε k 3 X ε 4 +( ε + +ε + By rewriting wehave k Y(ε 0 + k X ε k 3 X ε 4 +( ε +ε + ][ε + +( ε +ε + ] ( ε + +ε [ε + (0 ( ( By retaining only the terms up to the second degree of ε s we have Y (k Y+k Yε 0 k X ε k 3 X ε + k Yε + (3 k Yε The bias of the proposed estimator Bias ( E( Y Y[(k + δk C x + δk C x ] (4 The of this new estimator with two auxiliary variables ( E( Y Y [(k δk (C x +δk (C x +C y ] δk YC y (k ρ yx X +k 3 ρ yx X +δ(k C x X +C x k k 3 ρ x x X X +k 3 C x X (5

4 Mathematical Problems in Engineering The optimum values of k k andk 3 are given by k A[+δ(C x ] (A+C y δl + Aδ (C x k C yy(ρ yx ρ x x ρ yx [+δ(c x ] X (A + Cy δl + Aδ (C x +Cx k 3 C yy(ρ yx ρ x x ρ yx [+δ(c x ] X (A + Cy δl + Aδ (C x +Cx (6 δy C y L/ (A + C y δl [ + Aδ (Cx +Cx /(A+Cy δl] δ Y A(C x /(A+C y δl 4[+Aδ(Cx +Cx /(A+Cy δl] min ( [ + δ (Cx +Cx min ( / min ( ] δ Y (C x min ( / min ( 4[+δ(C x min ( / min ( ] The minimum of can be shown as min ( δy [4C y L Aδ(C x ] 4[A+C y δl + Aδ (C x ] (7 6 Comparison of with Some Existing Estimators We compared the of the proposed estimator with two auxiliaryvariablesgivenin(7withtheoftheestimator reported by Abu-Dayyeh et al [7] as given in (4 Kadilar and Cingi [8] as given in (8 Malik and Singh [9] as given in ( and the regression estimator as given in (7 as follows: min ( +M Y δ(c x M 4 (+M where Mδ(C x min ( / min ( M δ(c x min ( min ( Y δ(c x M >0 4 (+M >0 min ( < min ( +M < min ( (9 (30 min ( < min ( < min (y AD Proof min ( δy C y L A+δC y L min ( min ( δy C y L/A always +(δy C y L/A (/Y min (y AD + min (y AD /Y < min (y AD min ( min ( δy [4C y L Aδ(C x ] min ( 4[A+Cy δl + Aδ (C x +Cx ] δy C y L [A + Cy δl + Aδ (C x +Cx ] δ Y A(C x 4[A+Cy δl + Aδ (C x +Cx ] (8 7 Numerical Application in Engineering To examine the merits of the proposed estimator we considered four natural population datasets in agricultural biomedical and power engineering We used the following formula to calculate the percent of relative efficiency of different estimators: PRE (φ y AD (y AD (φ where φ or or or 00 (3 Population I (sourceinbiomedicalengineering[4] Y: number of placebo children X : number of paralytic polio cases in the placebo group X : number of paralytic polio cases in the not inoculated group N 34 Y 49 X 59 X 9 ρ yx 0736 ρ yx 06430 ρ x x 06837 C y 03 39and 070 Population II (source in agricultural engineering Y:cottonoutput X :theareaoftheplant

Mathematical Problems in Engineering 5 Table : and PRE values of different estimators about population I PRE (φ y AD n n/n y AD 0 094 0738607 0738607 0738607 076737 0433563 5 044 0390406 0390406 0390406 038409 097354 0 0588 0030 0030 0030 00907 0849 5 0735 00555 00555 00555 005057 009866 0 094 00 00 00 0305 7036 5 044 00 00 00 06 39 0 0588 00 00 00 0087 564 5 0735 00 00 00 0044 0747 Table : and PRE values of different estimators about population II PRE (φ y AD n n/n y AD 0 0056 70330 70330 70330 68867 607564 50 078 060505 060505 060505 06059 05897 90 0500 00094 00094 00094 00043 009985 30 07 0038536 0038536 0038536 003859 0038486 0 0056 00 00 00 00870 05955 50 078 00 00 00 033 00894 90 0500 00 00 00 0005 00343 30 07 00 00 00 0000 003 X : the proportion of good seed N 80 Y 3995 X 7398 X 38767 ρ yx 05630 ρ yx 0573 ρ x x 0589 C y 0480 0454and 03339 Population III (source in biomedical engineering [4] Y: weight measurement of children X : midarm circumference of children X : skull circumference of children N55 Y 708 X 69 X 5044 ρ yx 054 ρ yx 05 ρ x x 008 C y 069 007 0065 Population IV (source in power engineering Y: Electricity consumption by region in China in 00 X : Electricity consumption by region in China in 00 X : Electricity consumption by region in China in 000 N 30 Y 54609 X 4899 X 45354 ρ yx 09986 ρ yx 0993 ρ x x 09965 C y 0694 0679and 0659 3 Results and Discussion and PRE values of different estimators about population IcanbeseeninTable and PRE values of different estimators about populationiicanbeseenintable and PRE values of different estimators about populationiiicanbeseenintable3 and PRE values of different estimators about populationivcanbeseenintable4 The relative efficiency was studied based on the traditional regression- or ratio-type estimators in many literatures (Abu-Dayyeh et al [7] Haq and Shabbir [5] and Verma et al [6] However we studied the relative efficiency based on the higher efficient estimators using two auxiliary variables and found that the efficiency of the proposed estimator is higher than the estimators noted above under any conditions Under different sample sizes and different datasets we notice from the data given in Tables 3 and 4 that the proposed estimator of a finite population mean using twoauxiliaryvariablesisalwaysmoreefficientthanthe estimators y AD and Although the expressions of the estimators reported by Abu-Dayyeh et al Kadilar and Cingi and Malik and Singh are different we note from the comparative study above that the minimum values for the estimators reported by Abu-Dayyeh et al [7] Kadilar and Cingi [8] and Malik and Singh [9] are equal and have same expression The regression estimator has a smaller valuethanthethreeestimatorsnotedabovewithincrease in sample fraction the and PRE values of the proposed estimator decrease Therefore a smaller sampling fraction yields better results relative to and PRE values when compared to a larger sampling fraction Moreover with a large sampling fraction the efficiency differential among all estimators in the present study is very small Therefore it is suggested that our estimators be used with small sampling

6 Mathematical Problems in Engineering Table 3: and PRE values of different estimators about population III PRE (φ y AD n n/n y AD 0 08 05048093 05048093 05048093 05040335 050337 0 0364 0059867 0059867 0059867 00598039 005979094 30 0545 00857 00857 00857 00845 008096 40 077 0079088 0079088 0079088 007903 0078973 0 08 00 00 00 00056 0006 0 0364 00 00 00 00005 0004 30 0545 00 00 00 000097 00099 40 077 00 00 00 000044 000090 Table 4: and PRE values of different estimators about population IV PRE (φ y AD n n/n y AD 0 0333 30709 30709 30709 30557 76434 5 0500 05474 05474 05474 05437 95763 0 0667 670838 670838 670838 67083 69770 5 0833 3539 3539 3539 35387 3403 0 0333 00 00 00 00007 4358 5 0500 00 00 00 000035 06438 0 0667 00 00 00 0000 0650 5 0833 00 00 00 0000 0339 fractions From this viewpoint the proposed estimator can save survey cost In some sampling yields the sample fraction is not very large due to the irreversibility or the high cost of the test Then the accuracy of the proposed estimator is higher Haq and Shabbir [5] also reported on estimators of finite population mean using two auxiliary attributes and found that their values were reduced when sample size increased Therefore the findings of the present are consistent with that study 4 Conclusions In this paper we proposed the improved estimator of a finite population mean by utilizing information on two auxiliary variables in SRS Bias and expressions of the proposed estimator wereobtainedweclearlyprovedthatthe new estimator is always more efficient than the estimators reported by Abu-Dayyeh et al [7] Kadilar and Cingi [8] and Malik and Singh [9] as well as the regression estimator using two auxiliary variables These theoretical conditions are also satisfied by the results of four numerical examples in agricultural biomedical and power engineering It should be notedthatasmallersamplesizeyieldsbetterresultsrelativeto and PRE values when compared to a larger sample size Thus for use with small sample size the suggested estimator would be cost-saving in actual practice and are therefore recommended for efficient estimation of finite population mean Conflicts of Interest The author declares that they have no conflicts of interest Acknowledgments This work was supported by the National Natural Science Foundation of China (no 4605 and Natural Science Foundation of Inner Mongolia Autonomous Region of China (no 07MS(LH00 and the author gratefully acknowledges this support References [] H P Singh and G K Vishwakarma Modified exponential ratio and product estimators for finite population mean in double sampling Austrian Statistics vol36pp7 5 007 [] L K Grover and P Kaur An improved estimator of the finite population mean in simple random sampling Model Assisted Statistics and Applicationsvol6nopp47 550 [3] LKGroverandPKaur Animprovedexponentialestimator for finite population mean in simple random sampling using an auxiliary attribute Applied Mathematics and Computation vol 8 no 7 pp 3093 3099 0 [4] RSinghSMalikMKChaudharyHKVermaandAAdewara A general family of ratio-type estimators in systematic sampling Reliability and Statistical Studiesvol5pp 73 8 0 [5] H P Singh and R S Solanki Improved estimation of population mean in simple random sampling using information on auxiliary attribute Applied Mathematics and Computation vol 8 no 5 pp 7798 78 0 [6] S Gupta and J Shabbir On improvement in estimating the population mean in simple random sampling Applied Statisticsvol35no5-6pp559 566008 [7] W A Abu-Dayyeh M S Ahmed R A Ahmed and H A Muttlak Some estimators of a finite population mean using

Mathematical Problems in Engineering 7 auxiliary information Applied Mathematics and Computation vol 39 no -3 pp 87 98 003 [8] C Kadilar and H Cingi A new estimator using two auxiliary variables Applied Mathematics and Computation vol6no pp 90 908 005 [9] S Malik and R Singh An improved estimator using two auxiliary attributes Applied Mathematics and Computation vol9no3pp0983 098603 [0] P Sharma and R Singh A class of exponential ratio estimators of finite population mean using two auxiliary variables Pakistan Statistics and Operation Research vol no pp 905 [] S Muneer J Shabbir and A Khalil Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables Communications in Statistics Theory and Methodsvol46no5pp8 907 [] T J Rao On certain methods of improving ratio and regression estimators Communications in Statistics Theory and Methods vol 0 no 0 pp 335 3340 99 [3] H P Singh and M R Espejo On linear regression and ratioproduct estimation of a finite population mean the Royal Statistical Society Series D The Statistician vol5no pp 59 67 003 [4] S Choudhury and B K Singh A class of chain ratio-product type estimators with two auxiliary variables under double sampling scheme the Korean Statistical Society vol 4 no pp 47 56 0 [5] A Haq and J Shabbir An improved estimator of finite population mean when using two auxiliary attributes Applied Mathematics and Computationvol4pp4 404 [6] H K Verma P Sharma and R Singh Some families of estimators using two auxiliary variables in stratified random sampling Investigacion Operacionalvol36nopp40 50 05

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