New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables

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1 Journal of Mathematics and System Science 7 (017) doi: / / D DAVID PUBLISHING New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables Jambulingam Subramani and Master Ajith S Department of Statistics Ramanujan School of Mathematical sciences Pondicherry University Puducherry, , India id: drjsubramani@yahoo.co.in and ajith.master9@gmail.com Abstract This manuscript deals with new class of almost unbiased ratio cum product estimators for the estimation of population mean of the study variable by using the known values auxiliary variable. The bias and mean squared error of proposed estimators are obtained. An empirical study is carried out to assess the efficiency of proposed estimators over the existing estimators with the help of some known natural populations and it shows that the proposed estimators are almost unbiased and it perform better than the existing estimators. Keywords: Auxiliary variable, Bias, Mean squared error,natural populations, Ratio and Product estimators, Simple random sampling. 1. Introduction The efficiency of the estimators of the population parameters can be increased by using the prior information of the study characteristics. In literature several estimators exist with auxiliary variables involved. The commonly used thepopulation parameters of the auxiliary variables are mean, median, coefficient of variation, coefficient of skewness,coefficient of kurtosis etc. Ratio method of estimation is extensively used because of its computational simplicity and applicability.the correlation between study and auxiliary variables are negative the product method of estimation is used. Several researchers have directed their efforts towards to get efficient estimators of population mean. These estimators are biased but the percentage relative efficiency is better than that of simple random sampling, ratio and product estimators. For this reason, we consider the problem of estimation of population mean of study variable using known values of the auxiliary variable. So we have suggested new class modified ratio cum product estimators for estimating the population mean of the study variable. To know more abouthistorical developments of the estimation of population mean are referred to Cochran[1,], Subramani and Master Ajith [4,5], Murthy[7,8],Subramani[14], Subramani and kumarapandiyan[15,16,17], Upadhyaya and Singh[18], Yan and Tian[19] and the references cited there in. Consider a finite population U of size N consisting of UU 1, UU, UU 3. UU NN units. Each U i = (X i, Y i ),(i =1,,3...N) has a pair of values. Here Y be the study variable and X be the auxiliary variable which is correlated with Y. If yy = {yy 1, yy, yy yy nn }, and xx = {xx 1, xx, xx xx nn } be n sample values. Let yyand xx be the sample means of the study and auxiliary variables,ss yy = 1 NN (YY NN 1 ii=1 ii YY),SS xx = 1 NN (XX NN 1 ii=1 ii XX) and SS xxxx = 1 NN (YY NN 1 ii=1 ii YY)(XX ii XX) be the population variance and covariance of the study variable and auxiliary variable. Similarly the coefficient of variations and

2 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables coefficient of co-varianceof these variables are defined as CC yy = YY, CC xx = SS xx XX and CC xxxx = SS xxxx XXYY = ρρcc xx CC yy where is the correlation coefficient The simple random sample mean without replacement is used only when there is no additional information of the study variable is available,. In simple random sampling without replacement the estimator of population mean yy ssssss is an unbiased estimator for the population mean. The auxiliary variable and study variable are positively correlated,cochran [1]introduced ratio estimator for the estimation of population mean and it is given by YY RR = yy X = RRXX xx the bias and mean squared error of ratio estimator up to first order approximations are B(YY RR ) = δyy [CC xx ρρcc xx CC yy ] MSEYY RR = δyy CC yy + CC xx ρρccxx CC yy Where δ = 1 f n,ff = nn NN Some existing modified ratio estimators with their biases and mean squared errors are in table 1 Table 1: Bias and MSE of Existing modified ratio estimators Existing Estimators Constants Bias Variance/Mean squared Error YY MMMM1 = yy CC xxxx + ββ 1 CC xx xx + ββ 1 Yan and Tian [19] θθ 1 = CC xx XX CC xx XX + ββ 1 BYY MMMM1 = δyyθθ 1 CC xx θθ 1 ρc x C y MSEYY MMMM1 = δyy [CC yy + θθ 1 CCxx θθ 1 ρc x C y ] YY MMMM = yy ββ 1XX + CC xx ββ 1 xx + CC xx θθ = ββ 1 XX ββ 1 XX + CC xx BYY MMMM = δyyθθ CC xx θθ ρc x C y MSEYY MMMM = δyy [CC yy + θθ CCxx θθ ρc x C y ] YY MMMM3 = yy CC xxxx + ρρ CC xx xx + ρρ θθ 3 = CC xx XX CC xx XX + ρρ BYY MMMM3 = δyyθθ 3 CC xx θθ 3 ρc x C y MSEYY MMMM3 = δyy [CC yy + θθ 3 CCxx θθ 3 ρc x C y ] YY MMMM4 = yy ρρxx + CC xx ρρxx + CC xx θθ 4 = ρρxx ρρxx + CC xx BYY MMMM4 = δyyθθ 4 CC xx θθ 4 ρc x C y MSEYY MMMM4 = δyy [CC yy + θθ 4 CCxx θθ 4 ρc x C y ] YY MMMM5 = yy CC xxxx + ββ CC xx xx + ββ Upadhyaya and Singh [18] θθ 5 = CC xx XX CC xx XX + ββ BYY MMMM5 = δyyθθ 5 CC xx θθ 5 ρc x C y MSEYY MMMM5 = δyy [CC yy + θθ 5 CCxx θθ 5 ρc x C y ] YY MMMM6 = yy ββ XX + CC xx ββ xx + CC xx Upadhyaya and Singh [18] θθ 6 = ββ XX ββ XX + CC xx BYY MMMM6 = δyyθθ 6 CC xx θθ 6 ρc x C y MSEYY MMMM6 = δyy [CC yy + θθ 6 CCxx θθ 6 ρc x C y ] The auxiliary variable and study variable are negatively correlated, product estimator (Murthy[7]) is used,the bias and mean squared error of product estimator up to first order approximations are B(YY RR ) = δyy [ρρcc xx CC yy ] MSEYY RR = δyy CC yy + CC xx + ρρccxx CC yy

3 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables Existing Estimators YY MMMM1 = yy CC xxxx + ββ 1 CC xx XX + ββ 1 θθ 1 = YY MMMM = yy ββ 1xx + CC xx ββ 1 XX + CC xx θθ = Constants Bias Variance/Mean squared Error CC xx XX CC xx XX + ββ 1 BYY MMMM1 = δyyθθ 1 ρc x C y MSEYY MMMM1 = δyy [CC yy + θθ 1 CCxx + θθ 1 ρc x C y ] ββ 1 XX ββ 1 XX + CC xx BYY MMMM = δyyθθ ρc x C y MSEYY MMMM = δyy [CC yy + θθ CCxx + θθ ρc x C y ] YY MMMM3 = yy CC xxxx + ρρ CC xx XX + ρρ θθ 3 = CC xx XX CC xx XX + ρρ BYY MMMM3 = δyyθθ 3 ρc x C y MSEYY MMMM3 = δyy [CC yy + θθ 3 CCxx + θθ 3 ρc x C y ] YY MMMM4 = yy ρρxx + CC xx ρρxx + CC xx θθ 4 = YY MMMM5 = yy CC xxxx + ββ CC xx XX + ββ θθ 5 = YY MMMM6 = yy ββ xx + CC xx ββ XX + CC xx θθ 6 = ρρxx ρρxx + CC xx BYY MMMM4 = δyyθθ 4 ρc x C y MSEYY MMMM4 = δyy [CC yy + θθ 4 CCxx + θθ 4 ρc x C y ] CC xx XX CC xx XX + ββ BYY MMMM5 = δyyθθ 5 ρc x C y MSEYY MMMM3 = δyy [CC yy + θθ 5 CCxx + θθ 5 ρc x C y ] ββ XX ββ XX + CC xx BYY MMMM6 = δyyθθ 6 ρc x C y MSEYY MMMM = δyy [CC yy + θθ 6 CCxx + θθ 6 ρc x C y ] Some existing modified product estimators with their biases and mean squared errors are given in table Table : Bias and MSE of Existing modified product estimators. Suggested Class of Estimators In this section, new class of modified ratio cum product estimators for the population mean by using the known parameters of auxiliary variableis proposed and also derived the bias and the mean squared errors of the proposed estimators. The proposed estimators are given by YY PP1 = αα 1 λλ 1 yy CC xx XX+ββ 1 + (1 αα CC xx xx +ββ 1 )γγ 1 yy CC xx xx +ββ 1 1 CC xx XX+ββ 1 YY PP = αα λλ yy ββ 1XX+CC xx + (1 αα ββ 1 xx +CC )γγ yy ββ 1xx +CC xx xx ββ 1 XX+CC xx YY PP3 = αα 3 λλ 3 yy CC xx XX+ρρ + (1 αα CC xx xx +ρρ 3)γγ 3 yy CC xxxx +QQ rr CC xx XX+QQ rr YY PP4 = αα 4 λλ 4 yy ρρxx +CC xx + (1 αα ρρxx +CC 4 )γγ 4 yy ρρxx +CC xx xx ρρxx+cc xx YY PP5 = αα 5 λλ 5 yy CC xx XX+ββ + (1 αα CC xx xx +ββ 5 )γγ 5 yy CC xxxx +ββ CC xx XX+ββ YY PP6 = αα 6 λλ 6 yy ββ XX + CC xx ββ xx + CC xx + (1 αα 6 )γγ 6 yy ββ xx + CC xx ββ XX + CC xx Where λλ ii = γγ +aa ii CC ii =,i = 1,,3,4,5,6. Hereaa yy +bb ii CC ii sand bb ii s are constants. and CC yy are the yy population variance and coefficient of variation of study variable respectively. It is reasonable to assume that the values of and CC yy are known from the previous studies..1 The Bias and Mean Squared error of the Proposed Estimator

4 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables To obtain the bias and mean squared error of the proposed estimator, Consider ee 0 = yy YY, ee YY 1 = xx XX XX, θθ 1 = CC xx XX CC xx XX+ββ 1,θθ = ββ 1 XX ββ 1 XX+CC xx θθ 3 = CC xx XX CC xx XX+ρρ,θθ 4 = ρρxx ρρxx+cc xx,θθ 5 = CC xx XX CC xx XX+ββ,θθ 6 = ββ XX ββ XX+CC xx EE(ee 0 ) = EE(ee 1 ) = 0, EE(ee 0 ) = 1 ff YY CC nn yy, EE(ee 1 ) = 1 ff XX CC nn xx, EE(ee 0 ee 1 ) = 1 ff ρρcc nn xxcc yy Substitute the values of ee 0 and ee 1 in the proposed class of estimators and neglecting the higher order expressions, we get BBYY PPPP = EE(YY PPPP YY ) BBYY PPPP = 1 [YY (PP ii 1) + PP ii BBYY MMMMMM + BBYY MMMMMM + QQ ii BBYY MMMMMM BBYY MMMMMM ] WhereBYY MMMMMM = δyyθθ ii CC xx θθ ii ρc x C y, andbyy MMMMii = δyyθθ ii ρc x C y, i=1,,3,4, δ = 1 f n, θθ 1 = CC xx XX CC xx XX+ββ 1,θθ = ββ 1 XX ββ 1 XX+CC xx θθ 3 = CC xx XX CC xx XX+ρρ,θθ 4 = ρρxx ρρxx+cc xx,θθ 5 = Where PP ii = (αα ii λλ ii + (1 αα ii )γγ ii ) and QQ ii = (αα ii λλ ii (1 αα ii )γγ ii ) CC xx XX CC xx XX+ββ,θθ 6 = ββ XX ββ XX+CC xx Whereλλ ii and γγ ii are as defined above. If we assume that aa ii =0, bb ii = 0 and αα ii = 1 then the proposed estimators are exactly equal to the estimators given in Table 1.If aa ii =0, bb ii = 0 and αα ii = 0 then the proposed estimators are exactly equal to the estimators given in Table. If we assume that aa ii = BBYY MMMMMM, bb ii = BBYY MMMMMM, αα ii = 1 orαα ii = 0, then the proposed estimators are almost unbiased ratio estimators corresponding to the estimators given in Table 1 and. The detailed derivation of the mean squared errors are given in the appendix and the final expression is obtained with only first order approximation in the Taylor series expansion as, MMMMMMYY PPPP = EEYY PPPP YY MMMMMMYY PPPP = 1 4 [4YY (PP ii 1) + (PP ii + QQ ii ) MMMMMMYY MMMMMM + YYBBYY MMMMMM + (PP ii QQ ii ) MMMMMMYY MMMMMM + YYBBYY MMMMMM 4YY (PP ii + QQ ii )BBYY MMMMMM + (PP ii QQ ii )BBYY MMMMMM + PP ii QQ ii V(yy ssssss )] WhereBYY MMMMMM = δyyθθ ii CC xx θθ ii ρc x C y, BYY MMMMii = δyyθθ ii ρc x C y MSEYY MMMMii = δyy CC yy + θθ ii CCxx θθ ii ρc x C y MSEYY MMMMii = δyy CC yy + θθ ii CCxx + θθ ii ρc x C y andv(yy ssssss ) = δ = δδyy CC yy The optimal value of αα ii ss are determined by minimizing the MSE (YY pppp ) with respect to αα ii. For this differentiate MSE with respect to αα ii and equate to zero. = 0, and we get the value of αα ii, as αα ii

5 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables αα ii = Where λλ ii = YY (γγ ii 1)(γγ ii λλ ii ) + γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + YY λλ ii BBYY MMMMMM γγ ii BBYY MMMMMM λλ ii γγ ii V(yy ssssss ) YY (λλ ii γγ ii ) + λλ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM +aa ii CC yy, γγ ii = 3. Efficiency Comparison +bb ii CC yy,, i = 1,,3,4,5,6 If that aa ii = BBYY MMMMMM, bb ii = BBYY MMMMMM thenλλ ii = λλ ii γγ ii V(yy ssssss ) +BBYY MMMMMM CC yy, γγ ii = +BBYY MMMMMM CC yy,i = 1,,3,4,5,6. Substitute these values in the biased estimator and taking expectation we get the proposed estimators are almost unbiased.the efficiency comparison of the mean squared errorsof proposed estimatorsunder optimum conditions with that of the existing estimators are as follows 3.1 Comparison of Proposed estimators and Modified Ratio Estimator δyy CC yy + θθ ii CCxx θθ ii ρc x C y MMMMMMYY MMMMMM MMMMMMYY PPPP YY (αα ii λλ ii + (1 αα ii )γγ jj 1) + αα ii λλ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + (1 αα ii ) γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM YY αα ii λλ ii BBYY MMMMMM + (1 αα ii )γγ ii BBYY MMMMMM + (αα ii λλ ii (1 αα ii )γγ ii V(yy ssssss )) δyy CC yy + θθ ii CCxx θθ ii ρc x C y YY (αα ii λλ ii + (1 αα ii )γγ ii 1) + δyy CC yy (αα ii λλ ii + (1 αα ii )γγ ii ) + θθ CC xx 3αα ii λλ ii + (1 αα ii ) γγ ii αα ii λλ ii + θθθθcc xx CC yy (αα ii λλ ii (1 αα ii )γγ ii ) (αα ii λλ ii (1 αα ii ) γγ ii ) δδyy CC yy + θθ CC xx θθθθccxx CC yy YY (PP ii 1) + δδyy PP ii CC yy + θθ CC xx PP ii + (PP ii + QQ ii )(QQ ii 1) θθθθcc xx CC yy QQ ii (PP ii 1) δδδδδδcc xx CC yy (QQ ii (PP ii 1) 1) (PP ii 1) + δδ{cc yy PP ii 1 + θθ CC xx PP ii + (PP ii + QQ ii )(QQ ii 1) 1} ρρ (PP ii 1) + δδ[cc yy PP ii 1 + θθ CC xx PP ii + (PP ii + QQ ii )(PP ii 1) 1] δδδδcc xx CC yy (QQ ii (PP ii 1) 1) 3. Comparison of Proposed estimators and Modified Product Estimators MMMMMMYY MMMMMM MMMMMMYY PPPP

6 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables δyy CC yy + θθ ii CCxx + θθ ii ρc x C y YY (αα ii λλ ii + (1 αα ii )γγ jj 1) + αα ii λλ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + (1 αα ii ) γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM YY αα ii λλ ii BBYY MMMMMM + (1 αα ii )γγ ii BBYY MMMMMM + (αα ii λλ ii (1 αα ii )γγ ii V(yy ssssss )) δyy CC yy + θθ ii CCxx + θθ ii ρc x C y YY (αα ii λλ ii + (1 αα ii )γγ ii 1) + δyy CC yy (αα ii λλ ii + (1 αα ii )γγ ii ) + θθ CC xx 3αα ii λλ ii + (1 αα ii ) γγ ii αα ii λλ ii + θθθθcc xx CC yy (αα ii λλ ii (1 αα ii )γγ ii ) (αα ii λλ ii (1 αα ii ) γγ ii ) δδyy CC yy + θθ CC xx + θθθθccxx CC yy YY (PP ii 1) + δδyy PP ii CC yy + θθ CC xx PP ii + (PP ii + QQ ii )(PP ii 1) θθθθcc xx CC yy QQ ii (PP ii 1) δδδδδδcc xx CC yy (QQ ii (PP ii 1) + 1) (PP ii 1) + δδ{cc yy PP ii 1 + θθ CC xx PP ii + (PP ii + QQ ii )(QQ ii 1) 1} ρρ (PP ii 1) + δδ{cc yy PP ii 1 + θθ CC xx (PP ii (PP ii + QQ ii )(QQ ii 1) 1) δδδδcc xxxx (QQ ii (PP ii 1) + 1) 4. Numerical Study In this section we consider three natural populations,population 1 (Singh and Chaudhary [11] page 177),population (Khoshnevisan et.al.[6])and Population 3 (Cochran[] page 15) and are used to obtain the biases and mean squared errors and also used to compare the percentage relative efficiency of proposed estimator with that of the existing estimators.the computed values of constants and parameters of these populations are given below Table 3 : Parameters and Constants of Different Populations Constants Population 1 Population Population 3 N n YY XX SS xx CC xx CC yy ββ ββ ρρ θθ

7 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables θθ θθ θθ θθ θθ λλ λλ λλ λλ λλ λλ γγ γγ γγ γγ γγ γγ αα αα αα αα αα αα

8 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables Proposed Estimators Table 4:Bias and MSE of Proposed Estimators Population 1 Population Population 3 Bias MSE Bias MSE Bias MSE YY pp1 1.78E E E YY pp 7.11E E E YY pp3 4.6E E E YY pp4-1.60e E E YY pp5-1.07e E E YY pp6-1.78e E E Existing Estimators Table 5:Bias and MSE of Existing Estimators Population 1 Population Population 3 Bias MSE Bias MSE Bias MSE YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM YY MMMM Table 6: Percentage Relative Efficiency of Proposed Estimators Population 1 Population Population 3 Proposed Modified Modified Modified Modified Modified Modified Estimators Ratio Product Ratio Product Ratio Product YY pp YY pp YY pp YY pp YY pp YY pp

9 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables Conclusions In this paper we have suggested a new class of the modified ratio cum product estimators for finite population mean of the study variable Y with known parameters of the auxiliary variable. The biases and mean squared errors of the proposed estimators are obtained and compared with that of some existing modified ratio and modified product estimators.theoretically we have shown that the proposed estimator is always more efficient than other existing estimators under the optimum values of α i. and aa ii = BBYY MMMMMM, bb ii = BBYY MMMMMM. We have also studied the performance of the proposed estimators for certain known natural populations, it shows that the proposed estimator has less bias and mean squared error than all these existing estimators. That is the proposed estimator is more efficient than all these existing estimators. Hence we strongly recommended that the proposed estimator is more preferable than these existing estimators. References 1. Cochran W.G.(1940):The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce, The Journal of Agricultural Science, 30, Cochran, W. G. (1977):Sampling Techniques. Third Edition, Wiley Eastern Limited. 3. Jambulingam Subramani and Master Ajith (016):Improved Ratio cum Product Estimator with Known Coefficient of Variation insimple Random SamplingJ. Adv. Res. Appl. Math. Stat.; 1(), Jambulingam Subramani and Master Ajith (016):Modified Ratio cum Product Estimators for Estimation of Finite Population Mean with Known Correlation Coefficient Biom. Biostat. Int J, 4(6): Khoshnevisan M., Singh R., Chauhan P., Sawan N. and Smarandache F. (007): A general family of estimators for estimating population mean using known value of some population parameter(s), Far East Journal of Theoretical Statistics, Murthy, M.N. (1964): Product method of estimation. Sankhya A, 6, Murthy, M.N. (1967): Sampling theory and methods. Statistical Publishing Society, Calcutta, India 8. Rajesh Tailor and Balkishan Sharma (009): A Modified Ratio-cum-product estimator of finite population mean using known coefficient of variation and coefficient of kurtosis statistics in transition, July Vol. 10, No. 1, pp Ramkrishna S. Solanki, Housila P. Singh and Surya K. Pal (015): Improved ratio-type estimators of finite population variance using quartiles, Hacettepe Journal of Mathematics and Statistics Volume 44 (3) Singh, D. and Chaudhary, F.S. (1986):Theory and analysis of sample survey designs. New Age International Publisher 11. Singh, H. P. and Agnihotri, N.(008): A general procedure of estimating population mean using auxiliary information in sample surveys. Statistics in Transition- new series, 9, Singh, H.P., Tailor, R., Tailor, R. and Kakran, M.S.(004):Animproved estimator of population mean using power transformation. Journal of the Indian Society of Agricultural Statistics, 58(), 3-30,

10 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables 13. Sisodia B.V.S. and V.K. Dwivedi (1981): A modified ratio estimator using coefficient of variation of auxiliary variable, Jour. Ind. Soc. Agri. Stat., Vol. 33(1), Pp , 14. Subramani, J (013): "Generalized modified ratio estimator for estimation of finite population mean," Journal of Modern Applied Statistical Methods, vol.1, pp Subramani J.and G. Kumarapandiyan (01): Modified Ratio Estimators for Population Mean Using Function of Quartiles of Auxiliary Variable, Bonfring International Journal of Industrial Engineering and Management Science, Vol., No., 16. Subramani, J.,and Kumarapandiyan, (01) G. Variance estimation using quartiles and their functionsof an auxiliary variable, International Journal of Statistics and Applications (5), Subramani and G.Kumarapandiyan (01): A class of almost unbiased modified ratio estimators for population mean withknown population parameters;elixir Statistics Upadhyaya, L.N. and Singh, H.P(1999): Use of transformed auxiliary variable in estimating the finite population mean, Biometrical Journal 41 (5), , 19. Yan, Z. and Tian, B. (010): Ratio Method to the Mean Estimation Using Co-efficient of Skewness of Auxiliary Variable, ICICA 010, Part II, CCIS 106, pp

11 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables Bias and MSE of Proposed Estimators APPENDIX The proposed class of ratio cum product estimators with known parameters of auxiliary variables are YY PP1 = αα 1 λλ 1 yy CC xx XX+ββ 1 + (1 αα CC xx xx +ββ 1 )γγ 1 yy CC xx xx +ββ 1 1 CC xx XX+ββ 1 YY PP = αα λλ yy ββ 1XX+CC xx + (1 αα ββ 1 xx +CC )γγ yy ββ 1xx +CC xx xx ββ 1 XX+CC xx YY PP3 = αα 3 λλ 3 yy CC xx XX+ρρ + (1 αα CC xx xx +ρρ 3)γγ 3 yy CC xxxx +QQ rr CC xx XX+QQ rr YY PP4 = αα 4 λλ 4 yy ρρxx +CC xx + (1 αα ρρxx +CC 4 )γγ 4 yy ρρxx +CC xx xx ρρxx+cc xx YY PP5 = αα 5 λλ 5 yy CC xx XX+ββ + (1 αα CC xx xx +ββ 5 )γγ 5 yy CC xxxx +ββ CC xx XX+ββ YY PP6 = αα 6 λλ 6 yy ββ XX + CC xx ββ xx + CC xx + (1 αα 6 )γγ 6 yy ββ xx + CC xx ββ XX + CC xx Where λλ ii = γγ +aa ii CC ii =, i = 1,,3,4,5,6 yy +bb ii CC yy To obtain the bias and mean squared error of the proposed estimators, Consider,ee 0 = yy YY, ee YY 1 = xx XX, δ = 1 ff XX,ff = nn nn NN CC θθ 1 = xx XX ββ,θθ CC xx XX+ββ = 1 XX CC θθ 1 ββ 1 XX+CC 3 = xx XX,θθ ρρxx CC xx CC xx XX+ρρ 4 =,θθ ρρxx+cc 5 = xx XX ββ,θθ xx CC xx XX+ββ 6 = XX Where i = 1,,3,4,5,6 ββ XX+CC xx EE(ee 0 ) = EE(ee 1 ) = 0, EE(ee 0 ) = δyy CC yy, EE(ee 1 ) = δxx CC xx, EE(ee 0 ee 1 ) = δρρcc xx CC yy Substitute these values in YY pppp and neglecting the higher order expressions, we get YY pppp = αα ii λλ ii YY(1 + ee 0 )(1 + θθ ii ee 1 ) 1 + (1 αα ii )γγ ii YY(1 + ee 0 )(1 + θθ ii ee 1 ) = YYαα ii λλ ii (1 + ee 0 )(1 θθ ii ee 1 + θθ ii ee 1 ) + (1 αα ii )γγ ii (1 + ee 0 )(1 + θθ ii ee 1 ) = YYαα ii λλ ii (1 θθ ii ee 1 + θθ ii ee 1 + ee 0 θθ ii ee 1 ee 0 ) + (1 αα ii )γγ ii (1 + θθ ii ee 1 + ee 0 + θθ ii ee 0 ee 1 ) YY pppp YY = YYαα ii λλ ii 1 θθ ii ee 1 + θθ ii ee 1 + ee 0 θθ ii ee 1 ee 0 + (1 αα ii )γγ ii (1 + θθ ii ee 1 + ee 0 + θθ ii ee 0 ee 1 ) BBYY PPPP = EE(YY PPPP YY ) = EEYY(αα ii λλ ii (1 θθ ii ee 1 + θθ ii ee 1 + ee 0 θθ ii ee 1 ee 0 ) + (1 αα ii )γγ ii (1 + θθ ii ee 1 + ee 0 + θθ ii ee 0 ee 1 ) 1)} = δyyαα ii λλ ii (1 + θθ ii CC xx θθ ii ρρcc xx CC yy ) + (1 αα ii )γγ ii 1 + θθ ii ρρcc xx CC yy 1

12 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables BBYY PPPP = YY(αα ii λλ ii + (1 αα ii )γγ ii 1} + αα ii λλ ii BBYY MMMMMM + (1 αα ii )γγ ii BBYY MMMMMM Take a substitution PP ii = (αα ii λλ ii + (1 αα ii )γγ ii ) and QQ ii = (αα ii λλ ii (1 αα ii )γγ ii ) then BBYY PPPP = YY(PP ii 1} + (PP ii + QQ ii ) BBYY MMMMMM + (PP ii QQ ii ) BBYY MMMMMM BBYY PPPP = 1 [YY (PP ii 1) + PP ii BBYY MMMMMM + BBYY MMMMMM + QQ ii BBYY MMMMMM BBYY MMMMMM ] The mean squared errors of the proposed estimatorsare MMMMMMYY PPPP = EEYY PPPP YY = EE{YY(αα ii λλ ii (1 θθ ii ee 1 + θθ ee 1 + ee 0 θθ ii ee 1 ee 0 ) + (1 αα ii )γγ ii (1 + θθ ii ee 1 + ee 0 + θθ ii ee 0 ee 1 ) 1) } = EE YY (αα ii λλ ii 1 θθ ii ee 1 + θθ ii ee 1 + ee 0 θθ ii ee 1 ee 0 + (1 αα ii ) γγ ii (1 + θθ ii ee 1 + θθ ii ee 0 ee 1 ) αα ii λλ ii (1 αα ii )γγ ii 1 θθ ii ee 1 + θθ ii ee 1 + ee 0 θθ ii ee 1 ee 0 (1 + θθ ii ee 1 + ee 0 + θθ ii ee 0 ee 1 ) αα ii λλ ii 1 θθee 1 + θθ ii ee 1 + ee 0 θθ ii ee 1 ee 0 (1 αα ii )γγ ii (1 + θθ ii ee 1 + ee 0 + θθ ii ee 0 ee 1 ) = δyy αα ii λλ ii 1 + 3θθ ii CC xx + CC yy 4θθ ii ρρcc xx CC yy + (1 αα ii ) γγ ii 1 + θθ ii θθ CC xx + CC yy + 4θθ ii ρρcc xx CC yy + αα ii λλ ii (1 αα ii )γγ ii 1 + CC yy αα ii λλ ii 1 + θθ ii CC xx θθ ii ρρcc xx CC yy (1 αα ii )γγ ii 1 + θθ ii ρρcc xx CC yy MMMMMMYY PPPP = YY (αα ii λλ ii + (1 αα ii )γγ ii 1) + αα ii λλ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + (1 αα ii ) γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM YY αα ii λλ ii BBYY MMMMMM + (1 αα ii )γγ ii BBYY MMMMMM + (αα ii λλ ii (1 αα ii )γγ ii V(yy ssssss ) MMMMMMYY PPPP = YY (PP ii 1) + αα ii λλ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + (1 αα ii ) γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM YY αα ii λλ ii BBYY MMMMMM + (1 αα ii )γγ ii BBYY MMMMMM + (αα ii λλ ii (1 αα ii )γγ ii V(yy ssssss ) We take a substitution PP ii = (αα ii λλ ii + (1 αα ii )γγ ii ) and QQ ii = (αα ii λλ ii (1 αα ii )γγ ii ) then the MSE is MMMMMMYY PPPP = YY (PP ii 1) + (PP ii + QQ ii ) MMMMMMYY 4 MMMMMM + YYBBYY MMMMMM + (PP ii QQ ii ) MMMMMMYY 4 MMMMMM + YYBBYY MMMMMM YY (PP ii + QQ ii ) BBYY MMMMMM + (PP ii QQ ii ) BBYY MMMMMM + PP ii QQ ii V(yy 4 ssssss )] MMMMMMYY PPPP = 1 4 [4YY (PP ii 1) + (PP ii + QQ ii ) MMMMMMYY MMMMMM + YYBBYY MMMMMM + (PP ii QQ ii ) MMMMMMYY MMMMMM + YYBBYY MMMMMM 4YY (PP ii + QQ ii )BBYY MMMMMM + (PP ii QQ ii )BBYY MMMMMM + PP ii QQ ii V(yy ssssss )]

13 New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables WhereBYY MMMMMM = δyyθθ ii CC xx θθ ii ρc x C y,byy MMMMii = δyyθθ ii ρc x C y MSEYY MMMMii = δyy CC yy + θθ ii CCxx θθ ii ρc x C y MSEYY MMMMii = δyy CC yy + θθ ii CCxx + θθ ii ρc x C y andv(yy ssssss ) = δ = δδyy CC yy CC xx XX δ = 1 f, θθ n 1 =,θθ CC xx XX+ββ = 1 where, i = 1,,3,4,5,6 ββ 1 XX ββ 1 XX+CC xx θθ 3 = CC xx XX CC xx XX+ρρ,θθ 4 = ρρxx ρρxx+cc xx,θθ 5 = CC xx XX CC xx XX+ββ,θθ 6 = ββ XX ββ XX+CC xx The optimal value of αα ii is determined by minimize the MSE (YY pppp ) with respect to αα ii. For this differentiate MSE with respect to αα ii and equate to zero. ie, = 0, and we get the value of αα αα ii, as ii YY (αα ii λ i + (1 αα ii )γγ ii 1)(λ i γγ ii ) + αα ii λλ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM ) (1 αα ii )γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM ) YY λλ ii BBYY MMMMMM + γγ ii BBYY MMMMMM + (λλ ii (1 αα ii )γγ ii V(yy ssssss ) = 0 αα ii = (YY (γγ ii 1)(γγ ii λλ ii ) + γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + YY λλ ii BBYY MMMMMM γγ ii BBYY MMMMMM λλ ii γγ ii V(yy ssssss )) (YY (λλ ii γγ ii ) + λλ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM + γγ ii MMMMMMYY MMMMMM + YYBBYY MMMMMM Where δ = 1 f, i = 1,,3,4,5,6 n λλ ii γγ ii V(yy ssssss ))