Improvement in Estimating the Population Mean in Double Extreme Ranked Set Sampling

Transcription

1 International Mathematical Forum, 5, 010, no. 6, Improvement in Estimating the Population Mean in Double Extreme Ranked Set Sampling Amer Ibrahim Al-Omari Department of Mathematics, Faculty of Sciences, Al al-bayt University P.O. Box , Mafraq 5113, Jordan alomari Khalifa Jaber Collage of Business Administration, Al-Ain University of Science and Technology, P.O. Box 64141, Al-Ain, UAE Abstract In this paper, modified ratio estimators of the population mean are suggested using double extreme ranked set sampling DERSS method. The newly suggested estimators are compared with their counterparts suggested by Singh and Tailor 003 using simple random sampling SRS, and with extreme ranked set sampling ERSS methods. It is shown that, the DERSS estimators are approximately unbiased of the population mean. It is found that, DERSS is more efficient than SRS and ERSS for estimating the population mean based on the same number of measured units. Also, the mean square error MSE of the DERSS ratio estimators with negative values of the correlation coefficient is less than the MSE with positive values. Keywords: Ratio estimators, Ranked set sampling, Simple random sampling 1 Introduction The ranked set sampling method was first suggested by McIntyre 195 for estimating the population mean. Takahasi and Wakimoto 1968 introduced the mathematical theory of the RSS method. Samawi et al introduced ranked set sampling method based on extreme units for estimating the population mean. Samawi and Muttlak 1996 suggested used RSS to estimate the population ratio. Jemain and Al-Omari 006 suggested multistage

2 166 A. Ibrahim Al-Omari and Kh. Jaber median ranked set sampling MMRSS method for estimating the population mean. Jemain et al. 007 suggested multistage extreme ranked set sampling MERSS for estimating the population mean. For more about RSS see Al- Saleh and Al-Omari 00, Al-Omari and Jaber 008, Islam et al. 009, Mahdizadeh and Arghami 009, Al-Omari et al. 008, and Al-Omari et al Sampling Methods The RSS can be describes as follows: Step 1: Randomly select m units from the target population. Step : Allocate the m selected units as randomly as possible into m sets, each of size m. Step 3: Without yet knowing any values for the variable of interest, rank the units within each set with respect to variable of interest. This may be based on personal professional judgment or done with concomitant variable correlated with the variable of interest. Step 4: Choose a sample for actual quantification by including the smallest ranked unit in the first set, the second smallest ranked unit in the second set, the process is continues in this way until the largest ranked unit is selected from the last set. Step 5: Repeat Steps 1 through 4 for n cycles to obtain a sample of size mn. The extreme ranked set sampling ERSS method can be described as follows: Step 1: Select m random samples each of size m units from the population. Step : rank the units within each sample with respect to a variable of interest by visual inspection or any other cost free method. Step 3: If the sample size m is even, then select from the first m/ sets the smallest ranked unit X together with the associated Y and from the second m/ sets the largest ranked unit X together with the associated Y. If the sample size m is odd, select from the first m 1/ sets the smallest ranked unit X together with the associated Y and from the next set the median ranked unit of X together with the associatedy, and from the other m 1/ sets the largest ranked unit X together with the associatedy. This step yields m sets of extreme ranked samples each of size m bivariate units.

3 Improvement in estimating the population mean in DERSS 167 Step 4: The procedure is repeated n times to increase the sample size to nm units. The DERSS method can be described as follows: Step 1: Randomly select m 3 bivariate units from the target population, and divide these units randomly into m sets each of size m. Step : If the sample size m is even, then select from the first m / sets the smallest ranked unit X together with the associated Y and from the second m / sets the largest ranked unit X together with the associated Y. If the sample size m is odd, select from the first mm 1/ sets the smallest ranked unit X together with the associated Y and from the next m sets the median ranked unit of X together with the associated Y, and from the other mm 1/ sets the largest ranked unit X together with the associated Y. This step yields m sets of extreme ranked samples each of size m bivariate units. Step 3: Without doing any actual quantification for the units obtained in Step, if the sample size m is even, select from the first m/ sets the smallest ranked unit X together with the associated Y and from the second m/ sets the largest ranked unit X together with the associated Y. If the sample size m is odd, select from the first m 1/ sets the smallest ranked unit X together with the associated Y and from the next set the median ranked unit of X together with the associated Y, and from the other m 1/ sets the largest ranked unit X together with the associated Y. Step 4: The procedure is repeated n times to increase the sample size to nm units. 3 Estimation of the population mean Let X 1,Y 1, X,Y,..., X m,y m be a random sample of bivariate units with probability density function pdf fx, y, cumulative distribution function cdf F x, y, with means μ X, μ Y, variances σx, σy and correlation coefficient ρ. Let X 11,Y 11, X 1,Y 1,..., X 1m,Y 1m ; X 1,Y 1, X,Y,..., X m,y m ;...; X m1,y m1, X m,y m,..., X mm,y mm bem independent bivariate random samples each of size m. Let X i1,y i[1], Xi,Y i[],..., Xim,Y i[m] be the order statistics of Xi1,X i,..., X im and the judgment order of Y i1, Y i,..., Y im for i =1,,..., m. Then the RSS units are X 11,Y 1[1], X,Y [],..., Xmm,Y m[m].

4 168 A. Ibrahim Al-Omari and Kh. Jaber The estimator of the population mean μ using SRS is defined by X SRS = 1 m m X i with variance Var XSRS = σ m. The RSS estimator of the population mean μ is Ȳ RSS = 1 m Y i, with variance Var σ Ȳ RSS = m m 1 m m μi μ. 3.1 Using SRS Singh and Tailor 003 suggested a modified ratio estimator of the population mean μ Y using SRS as ˆμ Y SRSr = ȲSRS μx + ρ. 1 X SRS + ρ Using Taylor series expansion of ˆμ Y SRSr about μ X and μ Y, for the first order approximation, the estimator in can be written as ˆμ Y SRSr = Ȳ SRS μ Y XSRS μ X. This estimator is approximately unbiased, where E ˆμ Y SRSr = μ Y, with variance given by Var ˆμ Y SRSr = Var Ȳ SRS + H Var XSRS HCov XSRS, ȲSRS. 3 For simplification, using the following two relations Var Ȳ SRS = β Var 1 XSRS + m σ Y 1 ρ, and we have Cov XSRS, ȲSRS = βvar XSRS, where β = ρ σ Y σ X, MSE ˆμ Y SRSr = H β Var 1 XSRS + m σ Y 1 ρ. 4

5 Improvement in estimating the population mean in DERSS Using DERSS, X 1,Y,..., Xmm m[m],y denote the measured DERSSE. The If the sample size is even, let X11,Y 1[1] X m+ m,y m+ [m] DERSS ratio estimator of the population mean μ Y ˆμ Y DERSSEr = Ȳ DERSSE [1],..., X m 1,Y m [1], is given by X DERSSE + ρ, 5 where X DERSSE and Ȳ DERSSE are the sample means for X and Y, respectively using DERSS, where X DERSSE = 1 k Xi1 m m + Xim, i=k+1 with and Var 1 X DERSSE = σ X1 m + Xm σ, with ȲDERSSE = 1 k m Var Ȳ DERSSE Y i[1] + m Yi[m] i=k+1,k = m, 1 = σ Y [1] m + σ Y [m]., X 1,Y,..., X m 1 1,Y m 1, [1] If the sample size is odd, let X11,Y 1[1] [1] X m+1 m+1,y m+1, X [ m+1 ] m+3 m,y m+3,..., X [m] mm m[m],y denote the measured DERSSO. The DERSSO ratio estimator of the population mean μ Y is given by ˆμ Y DERSSOr = Ȳ DERSSO X DERSSO + ρ, 6 where X DERSSO = 1 l m Xi1 m + X l+1 m+1/ + X im,l = m 1, i=l+ with variance Var X DERSSO = m 1 m σ X1 + σ Xm + 1 m σ X m+1

6 170 A. Ibrahim Al-Omari and Kh. Jaber and ȲDERSSO = 1 l X m i[1] m + X l+1 [m+1/] + X i[m], i=l+ with variance Var ȲDERSSO m 1 = σ m Y [1] + 1 σ Y [m] + m σ Y [ m+1 ]. Using Taylor series expansion of ˆμ Y DERSSj,j = E,O about μ X and μ Y have ˆμ Y DERSSrj = ȲDERSSj H X DERSSj μ X X DERSSj μ X + H Ȳ X DERSSj μ X DERSSj μ Y we, 7 where j = E,O, and H = μ Y /. Take the expectation of 7 to obtain E ˆμ Y DERSSrj = μ Y + H Var X DERSSj Cov X DERSSj, Ȳ DERSSj, 8 and bias Bias ˆμ Y DERSSrj = H Var X DERSSj Cov X DERSSj, Ȳ DERSSj, 9 where Cov X DERSSj, Ȳ DERSSj = E X DERSSj μ X Ȳ DERSSj μ Y. For the first degree of approximation of Taylor series expansion, is given by ˆμ Y DERSSrj = Ȳ DERSSj H X DERSSj μ X. 10 The expectation of is E ˆμ Y DERSSrj = μ Y, indicate that the estimator is approximately unbiased. The variance of ˆμ Y DERSSr is given by Var ˆμ Y DERSSrj = Var Ȳ DERSSj + H Var X DERSSj HCov X DERSSj, Ȳ DERSSj. 11 By using Var Ȳ DERSSj = β Var 1 X DERSSj + m σ Y 1 ρ,

7 Improvement in estimating the population mean in DERSS 171 and Cov X DERSSj, Ȳ DERSSj = βvar X DERSSj, we have MSE ˆμ Y DERSSrj = H β Var 1 X DERSSj + m σ Y 1 ρ. 1 The efficiency of ˆμ Y DERSSrj with respect to ˆμ Y SRSr is defined as eff ˆμ Y DERSSrj, ˆμ Y SRSr = = MSE ˆμ Y SRSr MSE ˆμ Y DERSSrj [H β σ X +σ Y 1 ρ ] H β σ X1 +σ Xm +σ Y 1 ρ,mis even m[h β σx +σ Y 1 ρ ] ] H β [m 1 σ X1 +σ +mσ Xm Y 1 ρ +σ X m+1 13,mis odd 4 Simulation Study To compare the suggested estimators of the population mean using DERSS via SRS and ERSS, a simulation study is conducted where the ranking is performed on the variable X. The samples were generated from a bivariate normal distribution N μ X,μ Y,σX,σ Y,ρ with pdf [ ] 1 x μx x μx y μy y μy 1 f X,Y x, y = πσ X σ Y 1 ρ e 1 ρ ρ + σ X σ X σ Y σ Y. For simulations we considered μ X = 6, μ Y = 3, σx = σy = 1 and ρ = ±0.99,±0.90,±0.80,±0.70,±0.50 for m =, 3, 4, 5,6. The results are summarized in Table 1 and Table for DERSS and ERSS, respectively. Based on results in Tables 1 and, the following remarks can be concluded: 1. For all cases in Tables 1, a gain in efficiency is obtained using DERSS for estimating the population mean ˆμ Y.. ˆμ Y DERSSr is more efficient with negative values of the correlation coefficient than the positive values based on the same sample size. For example, with m = 4 for ρ =0.90, the efficiency is and for ρ = 0.90 the efficiency is The efficiency of the suggested estimators is increasing as the magnitude of the correlation coefficient increases. For example, for m = 3 with ρ =0.50, 0.70, 0.80, 0.90, 0.99 the efficiency values of ˆμ Y DERSSr are 1.06, 1.115, 1.6,1.64 and 3.056, respectively.

8 17 A. Ibrahim Al-Omari and Kh. Jaber Table 1: The efficiency and bias values of ˆμ DERSSr with respect to ˆμ Y SRSr for estimating the population mean of BN6, 3, 1, 1,ρ with m =, 3, 4, 5, 6. ρ m = m =3 m =4 m =5 m = Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS Efficiency Bias of DERSS Bias of SRS

9 Improvement in estimating the population mean in DERSS 173 Table : The efficiency and bias values of ˆμ ERSSr with respect to ˆμ YSRSr for estimating the population mean of BN6, 3, 1, 1,ρ with m =, 3, 4, 5, 6. ρ m = m =3 m =4 m =5 m = Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS Efficiency Bias of ERSS Bias of SRS

10 174 A. Ibrahim Al-Omari and Kh. Jaber 4. For several particular values of the sample size or correlation coefficient, the bias obtained using DERSS is less than its counterpart obtained using SRS. For example for m = 6 and ρ = 0.70, the Bias ˆμ Y DERSSr is while the Bias ˆμ Y SRSr is DERSS estimators of the population mean are more efficient than ERSS estimators for all cases considered in this study. As an example, for m = 5 and ρ =0.80, the efficiency values are 1.96 and using DERSS and ERSS, respectively. 5. Conclusion The ratio estimators for the population mean suggested in this paper using DERSS are approximately unbiased and more efficient than its counterparts obtained using SRS method and ERSS methods based on the same number of measured units. The DERSS ratio estimator is more efficient with negative values of the correlation coefficient than the positive values. References [1] Al-Omari, A.I., Jaber, K., and Al-Omari, A. Modified Ratio-Type Estimators of the Mean using Extreme Ranked Set Sampling. Journal of Mathematics and Statistics, 43, 008, [] Al-Omari, A.I., Ibrahim, K. and Jemain, A.A. New ratio estimators of the mean using simple random sampling and ranked set sampling methods. Revista Investigacin Operacional, 30, 009, [3] Al-Omari, A.I. and Jaber, K. Percentile double ranked set sampling. Journal of Mathematics and Statistics, 41, 008, [4] Al-Saleh, M.F. and Al-Omari, A.I. Multistage ranked set sampling. Journal of Statistical Planning and Inference, 00, 10: [5] Cochran, W.G. Sampling Technique. 3rd edition,wiley and Sons. Stateplace New York, [6] Islam, T., Shaibur, M.R., and Hossain, S.S. Effectivity of modified maximum likelihood estimators using selected ranked set sampling Data. Austrian Journal of Statistics, 38,009, [7] Jemain, A.A. and Al-Omari, A.I. Multistage median ranked set samples for estimating the population mean, Pakistan Journal of Statistics, 006,

11 Improvement in estimating the population mean in DERSS 175 [8] Jemain, A.A., Al-Omari, A.I., and Ibrahim, K. Multistage extreme ranked set samples for estimating the population mean. Journal of Statistical Theory and Applications, 64, 007, [9] Mahdizadeh, M., and Arghami, N.R. Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law. Journal of Statistical Computation and Simulation, 009, [10] McIntyre, G.A. A method for unbiased selective sampling using ranked sets, Australian Journal of Agricultural Research, 3 195, [11] Samawi. H.M. and Muttlak, H.A. Estimation of ratio using rank set sampling. The Biometrical Journal, , [1] Singh, H.P. and Tailor, R. Use of known correlation coefficient in estimating the finite population mean. Statistics in Transition, 6 003, [13] Takahasi K. and Wakimoto, K. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics, , Received: November, 009