Efficient Generalized Ratio-Product Type Estimators for Finite Population Mean with Ranked Set Sampling

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1 AUSTRIAN JOURNAL OF STATISTIS Volume 4 (013, Number 3, Efficient Generalized Ratio-Product Type Estimators for Finite Population Mean with Ranked Set Sampling V. L. Mowara Nitu Mehta (Ranka M. L. Sukhadia University, Udaipur, Rajasthan, India Abstract: In this paper we suggest two modified estimators of the population mean using the power transformation based on ranked set sampling (RSS. The first order approximation of the bias of the mean squared error of the proposed estimators are obtained. A generalized version of the suggested estimators by applying the power transformation is also presented. Theoretically, it is shown that these suggested estimators are more efficient than the estimators in simple rom sampling (SRS. A numerical illustration is also carried out to demonstrate the merits of the proposed estimators using RSS over the usual estimators in SRS. Zusammenfassung: In dieser Arbeit schlagen wir zwei modifizierte Schätzer für das Populationsmittel vor, indem wir die Power-Transformation basierend auf Ranked Set Sampling (RSS verwenden. Approximationen erster Ordnung für den Bias und den mittleren quadratischen Fehlers der vorgeschlagenen Schätzer werden erhalten. Eine verallgemeinerte Version der vorgeschlagenen Schätzer durch die Anwendung der Power-Transformation wird auch vorgestellt. Theoretisch wird gezeigt, dass diese vorgeschlagenen Schätzer effizienter sind als die Schätzer unter Simple Rom Sampling (SRS. Eine numerische Darstellung wird auch durchgeführt, um die Vorzüge dieser Schätzer unter RSS über die üblichen Schätzer unter SRS aufzuzeigen. Keywords: Ranked Set Sampling, Ratio Estimator, Power Transformation Estimator, Auxiliary Variable, oefficient of Variation, oefficient of Kurtosis. 1 Introduction The literature on ranked set sampling describes a great variety of techniques for using auxiliary information to obtain more efficient estimators. Ranked set sampling was first suggested by McIntyre (195 to increase the efficiency of estimator of population mean. Kadilar, Unyazici, ingi (009 used this technique to improve ratio estimator given by Prasad (1989. Here we shall propose two modified estimators of population mean using power transformation using RSS based on auxiliary variable. The classical ratio estimators given by ochran (1940 for estimating the population mean Y is defined as ( X y R = y, x where y is the sample mean for study variable y x, X are the sample mean population mean, respectively, for the auxiliary variable x.

2 138 Austrian Journal of Statistics, Vol. 4 (013, No. 3, When the population coefficient of variation x of the auxiliary variable x is known, Sisodia Dwivedi (1981 give a modified ratio estimator for Y as ( X + x y SD = y. (1 x + x Motivated by Sisodia Dwivedi (1981, H. P. Singh Kakran (1993 developed a ratio-type estimator for Y as ( X + β (x y SK = y, ( x + β (x where β (x is the known value of the coefficient of kurtosis of an auxiliary variable x. Utilizing the information on the coefficient of variation x the coefficient of kurtosis β (x of the auxiliary variable x, Upadhyaya Singh (1999 suggested the following ratio type estimators ( Xβ (x + x y UP 1 = y, (3 xβ (x + x ( Xx + β (x y UP = y. (4 x x + β (x By applying the power transformation on the Upadhyaya Singh (1999 estimators, H. P. Singh, Tailor, Singh, Kim (008 suggested modified estimators as ( α Xβ (x + x y UP 1(α = y (5 xβ (x + x ( δ Xx + β (x y UP (δ = y, (6 x x + β (x where α δ are suitably chosen scalars such that the mean squared errors of y UP 1(α y UP (δ are minimum. To the first degree of approximation, the mean squared error (MSE of these estimators are MSE(y R = 1 n Y ( y + x ρ yx y x, MSE(y SD = 1 n Y ( y + λ 1 x λ 1 ρ yx y x, (7 MSE(y SK = 1 n Y ( y + λ x λ ρ yx y x, (8 MSE(y UP 1 = 1 n Y ( y + γ 1 x γ 1 ρ yx y x, (9 MSE(y UP = 1 n Y ( y + γ x γ ρ yx y x, (10 MSE(y UP 1(α = 1 n Y ( y + ϕ 1α x ϕ 1 αρ yx y x, (11 MSE(y UP (δ = 1 n Y ( y + ϕ δ x ϕ δρ yx y x, (1

3 V.L. Mowara, Nitu Mehta 139 with y = S y Y, γ 1 = ϕ 1 = x = S x X, λ 1 = Xβ (x Xβ (x + x, γ = ϕ = X X + x, λ = X x X x + β (x, where ρ yx denotes the correlation coefficient between y x. X X + β (x, ρ yx = S yx S y S x, Ratio Estimator in Ranked Set Sampling In ranked set sampling (RSS m independent rom sets are chosen (each of size m the units in each set are selected with equal probability without replacement from a finite population of size N. The members of each rom set are ranked with respect to the characteristic of the study variable or auxiliary variable. Then, the smallest unit is selected from the first ordered set the second smallest unit is selected from the second ordered set. By this way, this procedure is continued until the unit with the largest rank is chosen from the m-th set. This cycle may be repeated r times, so mr = n units have been measured during this process. Thus, RSS SRS have equivalent sample sizes n for comparison of their biases efficiencies. When we rank on the auxiliary variable, let (y [i], x (i denote the i-th judgment ordering of the study variable the i-th perfect ordering for the auxiliary variable in the i-th set, where i = 1,..., m. Samawi Muttlak (1996 define the ratio estimator for the population mean as where y [n] = 1 n ( X y R,RSS = y [n], (13 x (n n y [i], x (n = 1 n are the ranked set sample means for the variables y x, respectively. To the first degree of approximation, the MSE of the estimator y R,RSS is given by (after ignoring the finite population correction factor ( MSE(y R,RSS = Y 1 mr ( y + x ρ yx y x (W y[i] W x(i, n x (i where with W x(i = 1 m r 1 X m τ x(i, W y[i] = 1 m r 1 Y τ x(i = µ x(i X, τ y[i] = µ y[i] Y. m τ y[i]

4 140 Austrian Journal of Statistics, Vol. 4 (013, No. 3, Suggested Estimators Based on Ranked Set Sampling Motivated by Sisodia Dwivedi (1981 we suggest a ratio-type estimator for Y using RSS, when the population coefficient of variation of the auxiliary variable x is known as ( X + x y RSS,MM1 = y [n]. (14 x (n + x To obtain the bias the MSE of this estimator we put y [n] = Y (1 + ε 0 x (n = X(1 + ε 1 so that E(ε 0 = E(ε 1 = 0 as E ( ( y [n] = Y E x(n = X under ranked set sampling. Moreover, var(ε 0 = E(ε 0 = var(y ( [n] Y = 1 1 mr Y Sy 1 m τy[i] m similarly where = θ y W y[i] with θ = 1 mr var(ε 1 = E(ε 1 = θ x W x(i cov(ε 0, ε 1 = E(ε 0 ε 1 = cov ( y [n], x (n W yx(i = 1 m r XY = θρ yx y x W yx(i, 1 XY = 1 XY ( 1 S yx 1 mr m m τ yx(i m τ yx(i with τ yx(i = (µ y[i] Y (µ x(i X. Further to validate the first degree of approximation, we assume that the sample size is large enough to get ε 0 ε 1 as small as possible such that the terms involving ε 0 or ε 1 in a degree greater than two are negligible. The bias the MSE of y RSS,MM1 are found next. Since bias(y RSS,MM1 = E(y RSS,MM1 Y y RSS,MM1 = Y (1 + ε 0 (1 + λ 1 ε 1 1 X, λ 1 = X + x we suppose λ 1 ε 1 < 1 so that (1 + λ 1 ε 1 1 is expable derive y RSS,MM1 = Y (1 + ε 0 { 1 λ 1 ε 1 + λ 1ε 1 + O(λ 1 ε 1 } Because E(ε 0 = E(ε 1 = 0 we have bias(y RSS,MM1 = Y ( λ 1E(ε 1 λ 1 E(ε 0 ε 1 = Y ( { } { } λ 1 θ x Wx(i λ1 θρyx y x W yx(i = Y ( θ(λ 1 x λ 1 ρ yx y x {λ 1W x(i λ 1 W yx(i }.

5 V.L. Mowara, Nitu Mehta 141 Now MSE(y RSS,MM1 = E(y RSS,MM1 Y = Y E ( ε 0 λ 1 ε 1 +λ 1ε 1 λ 1 ε 0 ε 1 = Y E ( ε 0+λ 1ε 1 λ 1 ε 0 ε 1 = Y ( θ y W y[i]+λ 1(θ x W x(i λ 1 ( θρyx y x W yx(i = Y ( θ{ y +λ 1 x λ 1 ρ yx y x } {W y[i]+λ 1W x(i λ 1 W yx(i } = Y ( θ{ y +λ 1 x λ 1 ρ yx y x } {W y[i] λ 1 W x(i }. (15 Motivated by H. P. Singh Kakran (1993 we suggest another new ratio estimator in ranked set sampling as ( X + β (x y RSS,MM = y [n]. (16 x (n + β (x Similarly, the bias the mean squared error are obtained, respectively, as bias(y RSS,MM = Y ( θ(λ x λ ρ yx y x {λ W x(i λ W yx(i } MSE(y RSS,MM = Y ( θ{ y + λ x λ ρ yx y x } {W y[i] λ W x(i }. (17 Motivated by Upadhyaya Singh (1999 we also proposed two more ratio type estimators considering both coefficients of variation kurtosis using ranked set sampling as ( Xβ (x + x y RSS,MM3 = y [n] (18 x (n β (x + x y RSS,MM4 = y [n] ( Xx + β (x x (n x + β (x The bias the MSE of y RSS,MM3 can be found as follows. Since bias(y RSS,MM3 = E(y RSS,MM3 Y y RSS,MM3 = Y (1 + ε 0 (1 + γ 1 ε 1 1, γ 1 =. (19 Xβ (x Xβ (x + x, we suppose that γ 1 ε 1 < 1 such that (1 + γ 1 ε 1 1 is expable get Therefore, y RSS,MM3 = Y (1 + ε 0 { 1 γ 1 ε 1 + γ 1ε 1 + O(γ 1 ε 1 }. bias(y RSS,MM3 = Y ( λ E(ε 1 λe(ε 0 ε 1 = Y ( { } { } γ1 θ x Wx(i γ1 θρyx y x W yx(i = Y ( θ(γ1 x γ 1 ρ yx y x {γ1w x(i γ 1 W yx(i }

6 14 Austrian Journal of Statistics, Vol. 4 (013, No. 3, MSE(y RSS,MM3 = E(y RSS,MM3 Y = Y E ( ε 0 γ 1 ε 1 +γ 1ε 1 γ 1 ε 0 ε 1 = Y E ( ε 0+γ 1ε 1 γ 1 ε 0 ε 1 = Y ( θ y W y[i]+λ (θ x W x(i λ ( θρ yx y x W yx(i = Y ( θ{ y +γ 1 x γ 1 ρ yx y x } {W y[i]+γ 1W x(i γ 1 W yx(i } = Y ( θ{ y +γ 1 x γ 1 ρ yx y x } {W y[i] γ 1 W x(i }. (0 Similarly, the bias mean squared error of the estimator y RSS,MM4 can be obtained, respectively, by changing the place of coefficient of kurtosis coefficient of variation, as bias(y RSS,MM4 = Y ( θ(γ x γ ρ yx y x {γ W x(i γ W yx(i } MSE(y RSS,MM4 = Y ( θ{ y + γ x γ ρ yx y x } {W y[i] γ W x(i }. (1 By applying the power transformation on y RSS,MM3 y RSS,MM4 given in (18 (19, we now propose generalized estimators as ( α Xβ (x + x y RSS,MM3(α = y [n] x (n β (x + x ( ( δ Xx + β (x y RSS,MM4(δ = y [n]. x (n x + β (x (3 The bias MSE of the estimator y RSS,MM3(α to the first degree of approximation, are obtained next. We have where bias(y RSS,MM3(α = E(y RSS,MM3(α Y y RSS,MM3(α = Y (1 + ε 0 (1 + ϕ 1 ε 1 α ( = Y (1 + ε 0 { 1 ϕ 1 αε 1 + } α(α + 1 ϕ 1ε 1 + O(ε 1 Thus ( bias(y RSS,MM3(α = Y ϕ α(α+1 { } 1 θ x Wx(i ϕ1 α { } θρ yx y x W yx(i = Y ( ϕ1 αθ { } (α+1 x ρ yx y x ϕ1 α { } (α+1wx(i W yx(i

7 V.L. Mowara, Nitu Mehta 143 MSE(y RSS,MM3(α = E(y RSS,MM3(α Y ( = Y E ε 0 ϕ 1 αε 1 +ϕ α(α+1 1 ε 1 ϕ 1 αε 0 ε 1 = Y E ( ε 0+ϕ 1α ε 1 ϕ 1 αε 0 ε 1 = Y ( θ y W y[i]+ϕ 1α (θ x W x(i ϕ 1 α ( θρ yx y x W yx(i = Y ( θ{ y +α ϕ 1 x αϕ 1 ρ yx y x } {W y[i] ϕ 1 αw x(i }. (4 Similarly, the bias mean squared error of the estimator y RSS,MM4(δ can be obtained, respectively, by changing the place of coefficient of kurtosis coefficient of variation, as bias(y RSS,MM4(δ = Y ( ϕ δθ { (δ+1 x ρ yx y x } ϕ δ { (δ+1w x(i W yx(i } MSE(y RSS,MM4(δ =Y ( θ{ y +δ ϕ x δϕ ρ yx y x } {W y[i] ϕ δw x(i }. (5 4 Optimality of α δ The optimum value of α to minimize the MSE of y RSS,MM3(α can easily be found as zero of its derivative, i.e. α MSE(y RSS,MM3(α = 0, which results in α = θρ xy x y W yx(i ϕ 1 (θx Wx(i because cov(x (n, y [n] = βvar ( x (n this is equivalent to Similarly, y α = ρ xy. (6 ϕ 1 x δ = ρ xy. (7 ϕ x After substituting (6 (7, respectively, in (4 (5, we obtain the minimum mean squared error of the proposed estimators as where K = ρ y x. y min MSE(y RSS,MM3(α = min MSE(y RSS,MM4(δ = Y ( θ y (1 ρ yx {W y[i] KW x(i },

8 144 Austrian Journal of Statistics, Vol. 4 (013, No. 3, Efficiency omparison On comparing (7 to (1 with (15, (17, (0, (1, (4 (5, respectively, we obtain 1. MSE(y SD MSE(y RSS,MM1 = A 1 0, where A 1 = [ W y[i] λ 1 W x(i ] MSE(y RSS,MM1 MSE(y SD. MSE(y SK MSE(y RSS,MM = A 0, where A = [ W y[i] λ W x(i ] MSE(y RSS,MM MSE(y SK 3. MSE(y UP 1 MSE(y RSS,MM3 = A 3 0, where A 3 = [ W y[i] γ 1 W x(i ] MSE(y RSS,MM3 MSE(y UP 1 4. MSE(y UP MSE(y RSS,MM4 = A 4 0, where A 4 = [ W y[i] γ W x(i ] MSE(y RSS,MM4 MSE(y UP 5. MSE(y UP 1(α MSE(y RSS,MM3(α = A 5 0, where A 5 = [ W y[i] ϕ 1 αw x(i ] MSE(y RSS,MM3(α MSE(y UP 1(α 6. MSE(y UP (δ MSE(y RSS,MM4(δ = A 6 0, where A 6 = [ W y[i] ϕ 1 δw x(i ] MSE(y RSS,MM4(δ MSE(y UP (δ It is easily seen that the MSE of the suggested estimators given in (14, (16, (18, (19, (, (3 are always smaller than those of the estimators given in (1 to (6, respectively, because A 1, A, A 3, A 4, A 5, A 6 are all non-negative values. As a result, the proposed generalized estimators y RSS,MM3(α y RSS,MM4(δ for the population mean using power transformation in ranked set sampling are more efficient than the usual estimators y UP 1(α y UP (δ. For α = δ = 1, the generalized estimators given in ( (3 turn to product type estimators. 6 Numerical Illustration To compare the efficiencies of the various estimators of our study, we take a population of size N = 50 (see page 1111 in the appendix of S. Singh, 003. The example considers the data of agricultural loans outsting of all operating banks in different states of the USA in 1997, where y is the real estate farm loans (study variable in 1000 $ x is the non-real estate loans (auxiliary variable in 1000 $. For the above population, the parameters are summarized as: Y = , X = , Y = , X = , S x = , S y = , x = 1.556, y = , ρ = , β (x = 1.915, λ 1 = , λ = , γ 1 = ϕ 1 = , γ = ϕ = 0.998, K = From this population we have taken 100 ranked set samples with size m = 4 number of cycles r = 3, so that n = mr = 1. For these 100 ranked set samples chosen, we have computed estimated MSE s of the proposed estimators y MM1,RSS, y MM,RSS, y MM3,RSS, y MM4,RSS, y RSS,MM3(α, y RSS,MM4(δ which are given in Table. Table 1

9 V.L. Mowara, Nitu Mehta 145 Table 1: Estimated MSE s of various estimators using SRS (Note that y UP 1(α = y UP (δ Estimator y SD y SK y UP 1 y UP y UP 1(α MSE shows the MSE s of the estimators y SD, y SK, y UP 1, y UP, y UP 1(α, y UP (δ, given in H. P. Singh et al. ( onclusion On comparing Table 1 with Table Table 3 for the 100 ranked set samples, we see that the MSE s of the proposed estimators are smaller than those of the estimators given by previous authors. As a result, all the proposed new ratio type estimators y MM1,RSS, y MM,RSS, y MM3,RSS, y MM4,RSS, y RSS,MM3(α y RSS,MM4(δ for the population mean using RSS are more efficient than the respective estimators y SD, y SK, y UP 1, y UP, y UP 1(α y UP (δ under SRS. Thus, if the coefficient of variation the coefficient of kurtosis are known for the auxiliary variable, then these proposed estimators are recommended for use in practice. Acknowledgements The authors are thankful to the referees for their valuable suggestion. References ochran, W. G. (1940. Some properties of estimators based on sampling scheme with varying probabilities. Australian Journal of Statistics, 17, -8. Kadilar,., Unyazici, Y., ingi, H. (009. Ratio estimator for the population mean using ranked set sampling. Statistical Papers, 50, McIntyre, G. A. (195. A method of unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research, 3, Prasad, B. (1989. Some improved ratio type estimators of population mean ratio in finite population sample surveys. ommunications in Statatistics, Theory Methods, 18, Samawi, H. M., Muttlak, H. A. (1996. Estimation of ratio using rank set sampling. Biometrical Journal, 38, Singh, H. P., Kakran, M. S. (1993. A modified ratio estimator using known coefficient of kurtosis of an auxiliary character. Singh, H. P., Tailor, R., Singh, S., Kim, J. M. (008. A modified estimator of population mean using power transformation. Statistical Papers, 49, Singh, S. (003. Advanced Sampling Theory with Applications, Volume II. Kluwer Academic Publishers.

10 146 Austrian Journal of Statistics, Vol. 4 (013, No. 3, Table : Estimated MSE s of different new estimators using RSS. Note that y RSS,MM3(α = y RSS,MM4(δ Estimator y RSS,MM1 y RSS,MM y RSS,MM3 y RSS,MM4 y RSS,MM3(α

11 V.L. Mowara, Nitu Mehta 147 Table 3: Estimated MSE s of different new estimators using RSS. Note that y RSS,MM3(α = y RSS,MM4(δ Estimator y RSS,MM1 y RSS,MM y RSS,MM3 y RSS,MM4 y RSS,MM3(α

12 148 Austrian Journal of Statistics, Vol. 4 (013, No. 3, Sisodia, B. V. S., Dwivedi, V. K. (1981. A modified ratio estimator using coefficient of variation of auxiliary variable. Journal of the Indian Society of Agricultural Statistics, 33, Upadhyaya, L. N., Singh, H. P. (1999. Use of transformed auxiliary variable in estimating the finite population mean. Biometrical Journal, 41, Authors address: V. L. Mowara Nitu Mehta (Ranka Department of Mathematics Statistics University ollege of Science M. L. Sukhadia University Udaipur Rajasthan India mowara vl@yahoo.co.in, nitumehta8@gmail.com