A family of product estimators for population mean in median ranked set sampling using single and double auxiliary variables

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1 A family of product estimators for population mean in median ranked set sampling using single and double auxiliary variables Azhar Mehmood Abbasi and Muhammad Yousaf Shahad Department of Statistics Quaid-i-Azam university, Islamabad,Pakistan Abstract In this article, we proposed a family of product estimators for estimating population mean in median ranked set samplingmrss using single and two auxiliary variables adopting the family of estimators proposed by Khoshnevisan, M., R. Singh, P. Chauhan, N. Sawan, and F. Smarandache 007. The expressions of bias and mean square error MSE are derived. A comprehensive simulation study has been conducted to compare the MSEs of the estimators.it is found that the estimators based on MRSS perform better than the estimators based on simple random sampling SRS. A real data set has also been used to illustrate the method. Keywords: Simple random samplingsrs, ranked set samplingrss,extreme ranked set samplingerss,median ranked set samplingmrss,bias and mean square errormsr. 1 Introduction Ranked set sampling RSS is an efficient sampling method alternative to SRS suggested by McIntyre 195 for estimating mean pasture and forage yield.samawi and Muttlak 1996 proposed ratio estimators of population mean in ranked set sampling RSS and showed that it is more efficient than conventional estimator based on SRS. Samawi et al introduced extreme ranked set sampling and Muttlak 1997 suggested median ranked setmrss sampling for estimating population mean. Muttlak 001 introduced regression estimators in extreme and median ranked set sampling and showed that these are more efficient than estimators based on SRS. In this paper, we used MRSS and SRS on the family of estimators suggested by Khoshnevisan, M., R. Singh, P. Chauhan, N. Sawan, and F. Smarandache 007 for estimating population mean of the study variable Y using two auxiliary variables X and Z. It came out that the estimators in MRSS perform better than the estimators based on SRS. The rest of the paper is organized as follows: In section, SRS, RSS and MRSS are described. Some important notations and results are included in section 3. In section 4 expressions for bias and MSEs are derived for single and two auxiliary variables. In section 5, simulation study is conducted for comparison of performance of SRS and MRSS in estimating population mean. Empirical study is conducted in section 6. Finally, concluding remarks are given in section 7. Corresponding author azharstat@gmail.com

2 Sampling methods This section delineates SRS, RSS and MRSS.1 Simple random samplingsrs we select a sample of size m units out of N total units in such a way that each unit in the target population has equal chance of being included in the sample. A sample may be selected with or without replacement. Sampling is said to be with replacement when from a population a sampling unit is drawn, observed and then returned to the population before another unit is drawn. On the other hand, sampling is said to be without replacement when from a population a sampling unit is choosen and not returned to the population before another unit is drawn.. Rank Set SamplingRSS For selecting a sample of size m, identify m units from the target population and partition them into m sets each of size m and rank the units within each set visually or any cost free method with respect to variable of interest. From the ith,,3,...m samples, select the ith smallest ranked unit for actual measurement. This completes one cycle for selecting m units. The whole procedure can be repeated r times, if needed, to get samples of size..3 Median ranked Set SamplingMRSS MRSS can be described as: draw m simple random sample each of size m from target population and partition them into m sets and rank the units within each set with respect to variable of interest by any cost free method. If m is odd, select from each set m + 1/th smallest ranked unit. If m is even, select from first m/ sets m/th smallest ranked units and from the last m/ sets select m + /th smallest ranked units. This completes one cycle for selecting m units. The whole procedure can be repeated r times, if needed, to get samples of size. 3 Some important notations and results X 1i, Y 1i,X i, Y i,...,x mi, Y mi,i = 1,, 3,..., m be the ith SRS of size m from bivariate normal distribution having parameters,µ Y,σ X,σ Y and ρ. We assume that ranking is performed on auxiliary variable X to estimate mean of study variable Y,say µ Y. If m is odd X 1m+1/j, Y 1m+1/]j, X m+1/j, Y m+1/]j,..., X mm+1/j, Y mm+1/]j, denotes MRSS in jth cycle. If m is even, X 1m/j, Y 1m/]j,X m/j, Y m/]j,..., X m/m/j, Y m/m/]j,x m+/m+/j, Y m+m+/]j,...,x mm+/j, Y mm+/]j,j = 1,, 3,..., r denotes MRSS in jth cycle. We will denote MRSS for even and odd m respectively by MRSS e and MRSS o. Let us suppose that EX i =,EY i = µ Y,EX ik = k,ey ik] = µ Y k],v arx i = σx, V ary i = σy, V arx ik = σxk,v ary ik] = σy k] and CovX ik, Y ik] = σ Xk Y k] The sample means and varinaces of X and Y under SRS are given by X SRS = Ȳ SRS = 1 1 X ij Y ij V ar X SRS = σ X V arȳsrs = σ Y The sample means and variances of X and Y under MRSS for odd m are given by

3 X MRSSo = 1 X im+1/j V ar X MRSSo = 1 σ Xm+1/ Ȳ MRSSo = 1 Y im+1/]j V arȳmrss o = 1 σ Y m+1/] The sample means and variances of X and Y under MRSS for even m are given by X MRSSe = 1 m/ X im/j + i= m+ X im+/j V ar X MRSSe = σ Xm/ Ȳ MRSSe = 1 m/ Y im/]j + i= m+ Y im+/]j V arȳmrss e = σ Y m/] Similarly, suppose that we have two auxiliary variable, say X and Z, and ranking is performed on auxiliary variable Z to estimate the mean of study variable Y, then we have the following notations: EX i =,EY i = µ Y,EZ i = µ Z,EX ik] = k],ey ik] = µ Y k],ez k = µ Zk, V arx i = σx V ary i = σy, V arx ik]=σxk],v ary ik] = σy k],v arz ik = σzk, CovX ik], Y ik] = σ Xk] Y k] CovX ik], Z ik = σ Xk] Z k,covy ik], Z ik = σ Yk] Z k The sample means and varinaces of X, Y and Z under SRS are given by X SRS = Ȳ SRS = Z SRS = X ij Y ij Z ij V ar X SRS = σ X V arȳsrs = σ Y V ar Z SRS = σ Z The sample means and variances of X, Y and Z under MRSS for odd m are given by X MRSSo = 1 X im+1/]j V ar X MRSSo = 1 σ Xm+1/] Ȳ MRSSo = 1 Y im+1/]j V arȳmrss o = 1 σ Y m+1/] Z MRSSo = 1 Z im+1/j V ar Z MRSSo = 1 σ Zm+1/ The sample means and variances of X and Y under MRSS for even m are given by X MRSSe = 1 m/ X im/]j + i= m+ X im+/]j V ar X MRSSe = σ Xm/]

4 Ȳ MRSSe = 1 m/ Y im/]j + i= m+ Y im+/]j V arȳmrss e = σ Y m/] Z MRSSe = 1 m/ Z im/j + i= m+ Z im+/j V ar Z MRSSe = σ Zm/ 4 Propsosed family of product estimators in MRSS 4.1 A family of product estimators in MRSSo using single auxiliary variable Khoshnevisan, M., R. Singh, P. Chauhan, N. Sawan, and F. Smarandache 007 proposed the family of estimator in SRS as given by ] g a + b ˆȲ SRS = ȲSRS αa X 1 SRS + b + 1 αa + b similarly, the family of product estimators in MRSSo when ranking is performed on auxiliary variable X is given by ] g a + b ˆȲ MRSSo = ȲMRSSo] αa X MRSSo + b + 1 αa + b where α and g are suitable constants and a and b are real number or function of some known parameters of the auxiliary variable X like coefficient of Kurkosis β X, coefficient of variation C X and standard deviaition σ X or correlation coefficient ρ. To find bias and MSE of, we write the error terms e 1 = X MRSSo and e = ȲMRSSo] µ Y µ Y such that, Ee 1 = Ee = 0, Ee 1 = C X, Ee = C Y ] and Ee 1 e = ρ XY ]C X C Y ] where C X = σ Xm+1/ / and C Y ] = σ Y m+1/] /µ Y are coefficient of variation and ρ XY ] is correlation between Xm + 1/ and Y m + 1/]. Note that here we droped finite population correction factorfpc as our population is infinitely large. The expression in terms of e s reduces to ˆȲ MRSSo = µ Y 1 + e 1 + αλe 1 g 3 where λ = a, Assuming αλe a +b 1 < 1, so that right hand size of Eq.3 is expandable. Expanding right side of Eq3 upto second power of e,s, we have ˆȲ MRSSo µ Y = µ Y e gαλe 1 gαλe 1 e + 1 gg + 1α λ e 1 Taking expectation on both sides of Eq4, we have Bias ˆȲ MRSSo = E ˆȲ MRSSo µ Y = µ Y gg + 1α λ C X gαλρ XY ] C X C Y ] Considering Eq4 upto first order of approximation,we have ˆȲ MRSSo µ Y = µy e gαλe

5 Squaring on both sides of Eq6 and then taking expection, we have MSE upto first order of approximation as given by MSE ˆȲ MRSSo = 1 σy m+1/] + g α λ σxm+1/ µ Y µ X gαλσ Xm+1/ Y m+1/] µ Y To obtain optimum value of α, we differentiate Eq7 with respect to α and put it equal to zero, we get α Opt = σ Xm+1/ Y m+1/] µ Y gλσ Xm+1/ Minimum.MSE ˆȲ MRSSo = σ Y m+1/] 1 ρ Xm+1/Ym+1/] 8 where ρ Xm+1/ Y m+1/] denotes correlation between X m+1/ and Y m+1/] It may be noted that Eq8 is same as MSE of linear regression estimator based on single auxiliary variable in MRSSo A family of product estimators in MRSSe using single auxiliary variable A family of product estimators in MRSSe when ranking is performed on auxiliary variable X is given by ] g a + b ˆȲ MRSSe = ȲMRSSe] αa X 9 MRSSe + b + 1 αa + b where α,g a and b are constants defined earlier To find bias and MSE of the estimator9,we write the error terms e 1 = X MRSSe and e = ȲMRSSe] µ Y µ Y 7 such that Ee 1 = Ee = 0,, Ee 1 = C X, Ee = C Y ] and Ee 1e = ρ XY ] C X C Y ] where C X = σ Xm// and C Y ] = σ Y m/]/µ Y are coefficient of variation and ρ XY ] is correlation between X m/ and Y m/]. Note that here we droped finite population correction factorfpc as our population is infinitely large. The expression 9 in terms of e s reduces to g ˆȲ MRSSe = µ Y 1 + e 1 + αλe 1 10 Expanding right side of Eq10 upto second power of e,s, we have ˆȲ MRSSe µ Y = µ Y e gαλe 1 gαλe 1e + 1 gg + 1α λ e 1 Taking expectation on both sides of Eq.11, we get Bias ˆȲ MRSSe = E ˆȲ MRSSe µ Y = µ Y gg + 1α λ CX gαλρc XCY ] Considering Eq11 upto first order of approximation,we have ˆȲ MRSSe µ Y = µy e gαλe Squaring on both sides of Eq13 and then taking expectation, we have MSE as given by MSE ˆȲ MRSSe = 1 σy m/] + g α λ σxm/ µ Y µ X gαλσ Xm/ Y m/] µ Y To obtain optimum value of α, we differentiate Eq14 with respect to α and put it equal to zero, we get α opt = σ Xm/ Y m/] µ Y gλσ Xm/ 14

6 Table 1: Some Product estimators from the family of estimators in MRSS Estimator α a b g ] XMRSS ˆȲ 1 = ȲMRSS] +ρ XY +ρ XY ] XMRSS ˆȲ = ȲMRSS] +C X +C X ] βx XMRSS ˆȲ 3 = ȲMRSS] +C X β X +C X ] XMRSS ˆȲ 4 = ȲMRSS] +σ X +σ X 1 1 ρ XY C X -1 1 β X C X σ X -1 Minimum.MSE ˆȲ MRSSe = σ Y m/] 1 ρ Xm/Ym/] 15 It may be noted that Eq15 is same as MSE of linear regression estimator based on single auxiliary variable in MRSSe. We can generate Several ratio and product estimators of population mean µ Y from Eq and Eq9 by taking different values of the constants a,b,α and g. Some product estimator that are members of the family of estimators are given in T able.1. These estimators could be used according to the availability of the information about auxiliary variable X. For instance, population correlation coefficient between X and Y and mean of the auxiliary variable X are known, one should choose the estimator ˆȲ 1 to estimate the mean of the study variable Y. 4. Proposed family of product estimator using two auxiliary variables 4..1 A family of product estimators using two auxiliary variables in MRSSo Again follwing Khoshnevisan et al. 007,the family of product estimator in SRS using two auxiliary variables X and Z is given by ] g1 ] g ˆȲ SRS a + b cµ Z + d = ȲSRS αa X SRS + b + 1 αa + b βc Z 16 SRS + d + 1 βcµ Z + d Similarly,the family of product estimator in MRSSo using two auxiliary variables X and Z assuming ranking is performed on Z is given by ] g1 ˆȲ MRSSo a + b cµ Z + d = ȲMRSSo] αa X MRSSo] + b + 1 αa + b βc Z MRSSo + d + 1 βcµ Z + d 17 To find bias and MSE of the estimator17,we write the error terms e 1 = X MRSS0], e = ȲMRSS0] µ Y µ Y and e 3 = Z MRSS0 µ Z µ Z ] g such that, Ee 1 = Ee = Ee 3 = 0, Ee 1 = C X], Y ], Ee = C Z Ee 3 = C Ee 1e = ρ X]Y ] C X] C Y ], Ee 1e 3 = ρ X]Z C X] C Z, Ee e 3 = ρ Y ]Z C Y ] C Z where CX = σ Xm+1/]/ and CY ] = σ Y m+1/] /µ Y CZ = σ Zm+1/]/µ Z are coefficient of variation and ρ X]Y ],ρ X]Z,ρ Y ]Z are correlation coefficient among Xm+1/],Y m+1/],zm+ 1/ The expanding right side of Eq.17 and retaining terms upto second orders of e s, we have ˆȲ MRSSo µ Y = µ Y g g +1λ β e 3 g λ βe 3 g 1 λ 1 αe 1+g 1 g λ 1 λ αβe 1e 3+g 1 g 1 +1λ 1α e 1 +e

7 g λ βe e 3 g 1 λ 1 αe 1e + g 1 g λ 1 λ αβe 1e e 3 18 Taking expectation on both sides of Eq.18,we have Bias ˆȲ MRSSo = µ Y g 1 g λ 1 λ αβρ X]ZCX]C Z g 1 λ 1 αρ X]Y ]CX]C Y ] + g 1 g 1 + 1λ 1α CX] +g g + 1λ β CZ g λ βρ Y ]Z Cy]C Z + g 1 g λ 1 λ αβv X]Y ]Z 19 a where λ 1 =, λ a +b = cµ Z cµ Z +d 1 and V X]Y ]Z = X MRSSo] ȲMRSSo] µ Y Z MRSSo µ Z µ Y µ Z E From Eq.18, we have MRSSo µ Y = µ Y e g λ βe 3 g 1 λ 1 αe 1 0 Taking square on both sides of Eq.0 and then taking expectation we have the expression for MSE of ȲMRSSo as given by MSE MRSSo = µ Y σ Y m+1/] µ Y +g λ β σ Zm+1/ µ Z +g1λ 1α σ Xm+1/ g µ λ β σ Y m+1/]zm+1/ X µ Y µ Z +g 1 g λ 1 λ αβ σ Xm+1/]Zm+1/ g 1 λ 1 α σ Xm+1/]Y m+1/] µ Z µ Y To find optimum values of the constants α and β, we differentiate Eq.1 with respect to α and β and equate them to zero. After solving two obtained equations for α and β, we get 1 α opt = σ Zm+1/ σ Xm+1/]Y m+1/] σ Y m+1/]zm+1/ σ Xm+1/]Zm+1/ g 1 λ 1 µ Y σ Xm+1/] σ Zm+1/ σ Xm+1/]Zm+1/ and β opt = µ Z σ X σ Y m+1/]zm+1/ σ Xm+1/Zm+1/ σ Xm+1/]Y m+1/] g λ µ Y σ Xm+1/] σ Zm+1/ σ Xm+1/]Zm+1/ Minimum.MSE ˆȲ MRSSo = σ Y m+1/] 1 RY m+1/].xm+1/]zm+1/ where RY m+1/].xm+1/]zm+1/ ρ Xm+1/]Y m+1/] + ρy m+1/]zm+1/ ρ Xm+1/]Y m+1/]ρ Xm+1/]Zm+1/ ρ Y m+1/]zm+1/ = 1 ρ Xm+1/]Zm+1/ is multiple correlation coefficient of Y m+1/] on X m+1/] and Z m+1/ in MRSSo. Note that minumum MSE of ˆȲ MRSSo is equal to MSE of regression estimator using two auxiliary variables.

8 4.. A family of product estimator using two auxiliary variables in MRSSe The proposed family of product estimators in MRSSe using two auxiliary variables X and Z assuming ranking is performed on Z ] g1 ˆȲ MRSSe a + b cµ Z + d = ȲMRSSe] αa X MRSSe] + b + 1 αa + b βc Z MRSSe + d + 1 βcµ Z + d To find bias and MSE of the estimator,we write the error terms e 1 = X MRSSe], e = ȲMRSSe] µ Y µ Y and e 3 = Z MRSSe µ Z µ Z ] g such that, Ee 1 = Ee = Ee 3 = 0, Ee 1 = C X], Ee = C Y ], Z Ee 3 = C Ee 1e = ρ X]Y ] C X] C Y ], Ee 1e 3 = ρ X]Z C X] C Z, Ee e 3 = ρ Y ]Z C Y ] C Z where CX = σ Xm/]/ and CY ] = σ Y m/]/µ Y CZ = σ Zm/]/µ Z are coefficient of variation and ρ X]Y ],ρ X]Z,ρ Y ]Z are correlation coefficient among Xm/],Y m/],zm/ Expanding right side of Eq upto second orders in terms of e, s, we have MRSSe µ Y = µ Y g g +1λ β e 3 g λ βe 3 g 1 λ 1 αe 1+g 1 g λ 1 λ αβe 1e 3+g 1 g 1 +1λ 1α e 1 +e g λ βe e 3 g 1 λ 1 αe 1e + g 1 g λ 1 λ αβe 1e e 3 3 By taking expectation on both sides of Eq.3, we get Bias ˆȲ MRSSe = µ Y g 1 g λ 1 λ αβρ X]ZCy]C Z g 1 λ 1 αρ X]Y ]CX]C Y ] + g 1 g 1 + 1λ 1α CX] +g g + 1λ β CZ g λ βρ Y ]Z Cy]C Z + g 1 g λ 1 λ αβv X]Y ]Z 4 a where λ 1 =, λ a +b = cµ Z cµ Z +d 1 and V X]Y ]Z = X MRSSo] ȲMRSSo] µ Y Z MRSSo µ Z µ Y µ Z E From Eq.3, we have ˆȲ MRSSe µ Y = µ Y e g λ βe 3 g 1 λ 1 βe 1 5 Taking square on both sides of Eq.5 and then taking expectation we have the expression for MSE of ȲMRSSe as given by MSE ˆȲ MRSSe = µ Y σ Y m/] + g µ λ β σ Zm/ + g Y µ 1λ 1α σ Xm+1/ g Z µ λ β σ Y m/]zm/ X µ Y µ Z +g 1 g λ 1 λ αβ σ Xm/]Zm/ g 1 λ 1 α σ Xm/]Y m/] µ Z µ Y To find optimum values of the constants α and β, we differentiate Eq.6 with respect to α and β and equate them to zero. After solving two equations for α and β, we get σzm/ σ Xm/]Y m/] σ Y m/]zm/ σ Xm/]Zm/ α opt = g 1 λ 1 µ Y σxm/] σ Zm/ σ Xm/]Zm/ 6 and β opt = µ Z σ X σ Y m+1/]zm/ σ Xm/Zm/ σ Xm/]Y m/] g λ µ Y σ Xm/] σ Zm/ σ Xm/]Zm/

9 Table : Some Product estimators from the family of estimator 17 in MRSS Estimator ] ] α β a b c d g 1 g ˆȲ 1 XMRSS] = ȲMRSS] +ρ XY ZMRSS +ρ Y Z +ρ XY µ Z +ρ Y Z ρ XY 1 ρ Y Z -1-1 ] ] ˆȲ XMRSS] = ȲMRSS] +C X ZMRSS +C Z +C X µ Z +C Z C X 1 C Z -1-1 ] ] ˆȲ 3 βx XMRSS] = ȲMRSS] +C X βz ZMRSS +C Z β Z µ Z +C Z 1 1 β X C X β Z C Z = ȲMRSS] β X +C X ] ] XMRSS] +σ X ZMRSS +σ Z +σ X µ Z +σ Z σ X 1 σ Z -1-1 Minimum.MSE ˆȲ MRSSe = σ Y m/] 1 RY m/].xm/]zm/ where RY m/].xm/]zm+1/ ρ Xm/]Y m/] + ρy m/]zm/ ρ Xm/]Y m/]ρ Xm/]Zm/ ρ Y m/]m/ = 1 ρ Xm/]Zm/ is multiple correlation coefficient of Y m/] on X m/] and Z m/ in MRSSe. Note that minumum MSE of ˆȲ MRSSe is equal to MSE of regression estimator using two auxiliary variables. It may be noted that several ratio and product estimators of population mean µ Y from Eq17 and Eq by taking different values of the constants a,b,c,d,α,β, g 1 and g. Some product estimator that are members of the family of estimators are given in T able.. 5 Simulation study In this section we formulate a computer simulation study to ascertain performance of the proposed estimators in MRSS and SRS.For this purpose we draw bivariate random sample of sizes m = 5, 7, 10 from bivariate normal distribution having parameters = 3,µ Y =,σ X = 1,σ Y = 1 and ρ = ±0.99, ±0.90, ±0.80, ±0.60. We assume that ranking is performed on the auxiliary varible X to estimate the mean of the study variable Y.The MSEs of the estimators ˆȲ i, i = 1,, 3, 4 against each sample size for different values of ρ are obtained after simulation. The MSEs so obtained are reported in T able.3 5. We also conducted simulation study for the case of two auxiliary variables X and Z which are correlated with the study variable Y. For this case we draw multivariate random sample of the sizes m = 5, 7, 10 from multivariate normal distribution having parameters = 3,µ Y =,µ Z = 5,σ X = 1,σ Y = 1 σz = 1 and for different values of ρ XZ assuming ρ XY = 0.90 and ρ Y Z = 0.90, while, in another case we took ρ XY = ρ Y Z and assumed ρ XZ = Suppose ranking is performed on Z to rank X and Y the MSEs of the estimators ˆȲ i, i = 1,, 3, 4 for different sample sizes are reported in T able.6 8. The following remarks based on the T able A gain in efficiency in term of minimum MSE is obtained by using MRSS for estimating population mean.. As expected,the product estimators are more precise when there exist negative correlation between the study variable and auxiliary variables. 3. Estimates becomes more precise as sample size gets large.

10 Table 3: MSEs of the product estimators in SRS* and MRSS using single auxiliary variable X for m = 5 ρ XY Estimator Method ˆȲ 1 SRS* MRSS ˆȲ SRS* MRSS ˆȲ 3 SRS* MRSS ˆȲ 4 SRS* MRSS Table 4: MSEs of the product estimators in SRS* and MRSS using single auxiliary variable X form m = 7 ρ XY Estimator Method ˆȲ 1 SRS* MRSS ˆȲ SRS* MRSS ˆȲ 3 SRS* MRSS ˆȲ 4 SRS* MRSS Table 5: MSEs of the product estimators in SRS* and MRSS using single auxiliary variable X for m = 10 ρ XY Estimator Method ˆȲ 1 SRS* MRSS ˆȲ SRS* MRSS ˆȲ 3 SRS* MRSS ˆȲ 4 SRS* MRSS

11 Table 6: MSEs of the product estimators in SRS* and MRSS using two auxiliary variable X and Z for m = 5 ρ XZ ρ XY = ρ Y Z Estimator Method ˆȲ 1 SRS* MRSS SRS* MRSS SRS* MRSS SRS* MRSS Table 7: MSEs of the product estimators in SRS* and MRSS using two auxiliary variable X and Z for m = 7 ρ XZ ρ XY = ρ Y Z Estimator Method ˆȲ 1 SRS* MRSS SRS* MRSS SRS* MRSS SRS* MRSS Table 8: MSEs of the product estimators in SRS* and MRSS using two auxiliary variable X and Z for m = 10 ρ XZ ρ XY = ρ Y Z Estimator Method ˆȲ 1 SRS* MRSS SRS* MRSS SRS* MRSS SRS* MRSS

12 Table 9: MSEs of the product estimators for estimating mean output Y of a factory using number of workers as single auxiliary variable X Estimator Method m = 5 m = 7 m = 8 ˆȲ 1 SRS* MRSS ˆȲ SRS* MRSS ˆȲ 3 SRS* MRSS ˆȲ 4 SRS* MRSS Table 10: MSEs of the product estimators for estimating mean output Y of a factory using two auxiliary variables viz number of workers X and fixed capital Z Estimator Method m = 5 m = 7 m = 8 1 SRS* MRSS SRS* MRSS SRS* MRSS SRS* MRSS Empirical study In this section we used real data set to ascertain the performance of the considered product estimators for estimating population mean of output of a factory in SRS and MRSS. To make the data linear, we took natural log of study variable-output of a factoryy and two auxiliary variable viz.number of workers X and fixed capital Z, for details see Murthy et al The population parameters are given by N = 80, = 5.338, µ Y = , µ Z = , σx = , σ Y = 0.164, σ Z = , β X = 1.509, β Z = , ρ XY = ,ρ XZ = , ρ Y Z = The MSEs of the proposed estimators ˆȲ i and ˆȲ i i = 1,, 3, 4 are obtained after simulation and are given in T able The estimators ˆȲ 1 and ˆȲ 1 generally performs better for the case of single and two auxiliary variables respectively in this empirical study. 7 Conclusion In this paper, we presented the method of estimating population mean using product estimators under single and two auxiliary variables in MRSS by adopting the family of estimators proposed by Khoshnevisan, M., R. Singh, P. Chauhan, N. Sawan, and F. Smarandache 007. We used Monte Carlo simulation to compare the performance of SRS and MRSS for estimating population mean. From T ables3 5, it is found that the estimators ˆȲ 1 generally performs better than other estimators

13 when there is negative high correlation between X and Y other wise the estimator ˆȲ 4 performs better among other considered estimators for the case of single auxiliary variable. In case of two auxiliary variables, it can be seen from T able6 8 that the estimator ˆȲ 4 outperforms than others estimators. It is, therefore, recommended to use MRSS than SRS to get more efficient estimates of a population mean. Acknowledgments The first author is thankful to the editor R.James Knaub for his constructive comments and suggestions to bring the paper in present form. References Khoshnevisan, M., R. Singh, P. Chauhan, N. Sawan, and F. Smarandache 007. A general family of estimators for estimating population mean using known value of some population parameter s. arxiv preprint math/ McIntyre, G A method for unbiased selective sampling, using ranked sets. Crop and Pasture Science 3 4, Murthy, M. N. et al Sampling theory and methods. Sampling theory and methods.. Muttlak, H Median ranked set sampling. Journal of Applied Statistical Science 6 4, Muttlak, H. A Regression estimators in extreme and median ranked set samples. Journal of Applied Statistics 8 8, Samawi, H. M., M. S. Ahmed, and W. Abu-Dayyeh Estimating the population mean using extreme ranked set sampling. Biometrical Journal 38 5, Samawi, H. M. and H. A. Muttlak Estimation of ratio using rank set sampling. Biometrical Journal 38 6,