In this paper we have suggested an improved class of ratio type estimators in estimating the finite population mean when information on minimum and maximum values of the auxiliary variable is known. The properties of the suggested class of estimators in terms of bias and mean square error are obtained up to first order of approximation. Two data sets are used for efficiency comparisons

Daraz, Umer

Khan, Hina

Shabbir, Javid

English

Digital Commons@Wayne State University

Journal of Modern Applied StatisticalMethodsVolume 17 | Issue 1 Article 207-19-2018Estimation of finite population mean by usingminimum and maximum values in stratifiedrandom samplingUmer DarazQuaid-i-Azam University, umerdaraz09@yahoo.comJavid ShabbirQuaid-i-Azam University, javidshabbir@gmail.comHina KhanGovernment College University, hinakhan@gcu.edu.pkFollow this and additional works at: https://digitalcommons.wayne.edu/jmasmPart of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and theStatistical Theory CommonsThis Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted forinclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState.Recommended CitationDaraz, Umer; Shabbir, Javid; and Khan, Hina (2018) "Estimation of finite population mean by using minimum and maximum valuesin stratified random sampling," Journal of Modern Applied Statistical Methods: Vol. 17 : Iss. 1 , Article 20.DOI: 10.22237/jmasm/1532007537Available at: https://digitalcommons.wayne.edu/jmasm/vol17/iss1/20Estimation of finite population mean by using minimum and maximumvalues in stratified random samplingCover Page FootnoteDear Editor, We have revised the manuscript as referees have suggested.This regular article is available in Journal of Modern Applied Statistical Methods: https://digitalcommons.wayne.edu/jmasm/vol17/iss1/20Journal of Modern Applied Statistical Methods May 2018, Vol. 17, No. 1, eP2548 doi: 10.22237/jmasm/1532007537  Copyright © 2018 JMASM, Inc. ISSN 1538 − 9472    doi: 10.22237/jmasm/1532007537 | Accepted: May 24, 2017; Published: July 19, 2018. Correspondence: Javid Shabbir, javidshabbir@gmail.com    2 Estimation of Finite Population Mean by Using Minimum and Maximum Values in Stratified Random Sampling Ummer Draz Quaid-i-Azam University Islamabad, Pakistan Javid Shabbir Quaid-i-Azam University Islamabad, Pakistan Hina Khan Government College University Lahore, Pakistan   In this paper we have suggested an improved class of ratio type estimators in estimating the finite population mean when information on minimum and maximum values of the auxiliary variable is known. The properties of the suggested class of estimators in terms of bias and mean square error are obtained up to first order of approximation. Two data sets are used for efficiency comparisons.  Keywords: Minimum and maximum values, stratified random sampling, mean squared error, efficiency  Introduction In survey sampling, auxiliary information is frequently used in various forms to increase the precision of estimators by taking advantage of the correlation between the study and auxiliary variables. In the proposed class of estimators, the minimum and maximum values of the auxiliary variable are used.  Mohanty and Sahoo (1995), Khan and Shabbir (2013), Khan (2015), Walia, Kaur, and Sharma (2015), and Cekim and Cingi (2016) proposed some estimators for the estimation of finite population mean under minimum and maximum values using known information of the auxiliary variable. Consider a finite population U = (U1, U2, U3,…, UN) of size N which is divided into L strata, each of size Nh (h = 1, 2,…, L), such that 1LhhN N== . Let Yhi and Xhi be the values of the study variable and the auxiliary variable, respectively, in the hth stratum of the ith unit, i = 1, 2,…, Nh. We select a sample of size nh from the hth stratum by using simple random sampling without replacement. Let DRAZ ET AL 3  1 11 1andh hN Nh hi h hii ih hY Y X XN N= == =     be the population means of the study variable Y and the auxiliary variable X in the hth stratum corresponding to the population means   1 11 1andL Lh h h hh hh hY W Y X W XN N= == =     respectively, where Wh = Nh/N is the known stratum weight. Let   ( ) ( )2 22 21 1and1 1h hN Nhi h hi hi iyh xhh hY Y X XS SN N= =− −= =− −    be the population variances in the hth stratum. Let Cyh = Syh/Y̅h and Cxh = Sxh/X̅h be the population coefficients of variations in the hth stratum. Let Cyxh = ρyxhCyhCxh be the population co-variance between the study variable (Y) and the auxiliary variable (X) in the hth stratum. Let   1 11 1andh hn nh hi h hii ih hy Y x Xn n= == =     be the sample means of the study variable (Y) and the auxiliary variable (X) in the hth stratum. The usual unbiased estimator for the population mean Y̅ is given by   st1Lh hhy W y==   (1)  The variance of y̅st is given by   ( ) 2 2 2st1VLh h h yhhy W Y C==   (2)  where  ESTIMATION OF FINITE POPULATION MEAN 4  1 1hh hn N = −     The usual ratio estimator for Y̅ in stratified random sampling is given by   Rs1Lhh hh hXy W yx= =      (3)  The bias and mean square error of y̅Rs in stratified random sampling are given by   ( ) ( )2Rs1BiasLh h h xh yshhy W Y C C= −   (4)  and   ( ) ( )2 2 2 2Rs1MSE 2Lh h h yh xh yshhy W Y C C C= + −   (5)  The separate linear regression estimator y̅lrs of the population mean Y̅ is given by   ( )lrs1ˆLh h yxh h hhy W y X x= = + −    (6)  where   2ˆ yxhyxhxhss =    is the sample regression coefficient whose population regression coefficient is βyxh. Expressions for bias and mean square error of y̅lrs are given by   ( ) 12lrs 031BiasLhh h yxh h xh hh yxhy W X C  =  −      (7)  and DRAZ ET AL 5  ( ) ( )2 2 2 2lrs1MSE 1Lh h h yh yxhhy W Y C = −   (8)  where   2 202 20rshrsh r sh h =    with   ( ) ( )21 for 1,2,3, ,1hrNhi h hi hirshhY Y X Xh LN =− −= =−   Mohanty and Sahoo (1995) proposed two estimators by using known information of minimum and maximum values of the auxiliary variable. The transformations are given by   mM mhi hhh hX XvX X+=+  (9)  and   MM mhi hhh hX xzX X+=+  (10)  where Xmh and XMh are the minimum and maximum values of the auxiliary variable in the hth stratum. In stratified sampling, the ratio estimators are given by   1s1Lhh h hh hVy W yv= =      (11)  and  ESTIMATION OF FINITE POPULATION MEAN 6  2s1Lhh h hh hZy W yz= =      (12)  Expressions for biases and mean squared errors of these estimators are given by   ( ) ( )2 21s 1 11BiasLh h h h h xh h yxhhy W Y t C t C= −   (13)   ( ) ( )2 22s 2 21BiasLh h h h h xh h yxhhy W Y t C t C= −   (14)  and   ( ) ( )2 2 2 2 21s 1 11MSE 2Lh h h h yh h yxh h xhhy W Y C t C t C= − +   (15)   ( ) ( )2 2 2 2 22s 2 21MSE 2Lh h h h yh h yxh h xhhy W Y C t C t C= − +   (16)  where   1 2m Mandh hh hh h h hX Xt tX X X X= =+ +   Walia et al. (2015) proposed some estimators by using known information on minimum and maximum values of the auxiliary variable. The transformation is given by   M1mhh hihXz XX= +   (17)  Walia et al. (2015) defined two estimators:   1m1s1 1Lhh hh hZy W yz= =      (18) DRAZ ET AL 7 and   1 1m2s1 1 1Lh z hh hh h z hZ Cy W yz C= +=  +    (19)  where   11M1mz h xhz hhhhhS SCXZXX= =+   Expressions for biases and mean square errors of these estimators, up to first order approximation, are given by   ( ) ( )2 2m1s 1 11BiasLh h h h xh h yxhhy W Y C C C = −   (20)   ( ) ( )2 2m2s 2 21BiasLh h h h xh h yxhhy W Y C C C = −   (21)  and   ( ) ( )2 2 2 2 21 1 11MSE 2Lm s h h h yh h yxh h xhhy W Y C C C C = − +   (22)   ( ) ( )2 2 2 2 22 2 21MSE 2Lm s h h h yh h yxh h xhhy W Y C C C C = − +   (23)  where   Mm1 2 2MMmmandhh hhhh hhhhh xhhhXX XXXC CX XX X SX X +  = = + + +    ESTIMATION OF FINITE POPULATION MEAN 8 Proposed Estimator Motivated by Koyuncu (2012), an improved class of estimators is proposed for estimating the finite population mean using known information on minimum and maximum values of the auxiliary variable under stratified random sampling. The proposed estimator is given by   ( )( )1 2DS 1 21exp2Lh hh hh h h hh h h h h hX xX Xy W k y kx x X x j =   −    = +     + +            (24)  where kih, i = 1, 2, are unknown constants whose values are to be determined such that the bias and mean square error are minimum, jh = XMh – Xmh, and α1, α2 are scalar quantities which contain the values (0, –1, 1). From (24), we obtain the different classes of estimators which are given in Table 1.   Table 1. Some classes of estimators in stratified random sampling  Subsets of y̅DS α1 α2       DS1 1 2=1= +Lhh h h h hhhxy W k y k LX 0 –1         DS2 1 2=1= +Lhh h h h hhhxy W k y k LX –1 0       DS3 1 2=1= +Lhh h h h hhhXy W k y k Lx 0 1         DS4 1 2=1= +Lhh h h h hhhXy W k y k Lx 1 0             DS5 1 2=1= +Lh hh h h h hhh hX xy W k y k Lx X 1 –1             DS6 1 2=1= +Lh hh h h h hhh hx Xy W k y k LX x –1 1             DS7 1 2=1= +Lh hh h h h hhh hX Xy W k y k Lx x 1 1             DS8 1 2=1= +Lh hh h h h hhh hx xy W k y k LX X  –1 –1  DRAZ ET AL 9 where   ( )( )exp2h hhh h hX xLX x j −=  + +     To obtain the properties of the proposed estimator, we define the following error terms: Let   0 1andh h h hh hh hy Y x Xe eY X   − −= =         such that E(eih) = 0 for i = 0, 1, ( )2 20E h h yhe C= , ( )2 21E h h xhe C= , and ( )0 1E h h h yxhe e C= . Rewriting (24) in terms of errors, we have   ( )( ) ( )1 211 1 1DS 1 0 1 2 1 111 1 1 exp 12 2Lh h hh h h h h h h hhg e gy W k Y e e k e e −− −= −   = + + + + +           where   1M mhhh h hXgX X X=+ −   Up to first order of approximation,   ( )( )1DS 1 0 1 1121 22 1 1 1 11 0 1 122 221 2 1 12 1 2 112132 8 2 21312 2 8 2Lhh h h h h hhh h hh h hh h hh h hgy Y W Y k Y e eg g ge e eg g gk e e  =   −  − + + − +   +   + + + − +        +   + − + + + +         (25)  Using (25), the bias of y̅DS, up to first order of approximation, is given by ESTIMATION OF FINITE POPULATION MEAN 10  ( ) ( )DS 1 21BiasLh h h h h h hhy W Y k Y D k G= − + +   (26)  where    ( )2 1 12 1 1 1 111312 8 2 2h h hh h xh h yxhg g gD C C    +   = + + + − +           and   ( )2 2 22 2 1 1 1312 8 2h hh h xhg gG C  + = + + +        By squaring and taking expectations on both sides of (25), obtain the mean square error up to first order of approximation, which is given by   ( ) ()2 2 2 2 2 2DS 1 2 112 1 2MSE 22 2Lh h h h h h h h h hhh h h h h h hy W Y Y k A k B Y k DY k G Y k k F= + + −− +  (27)  where   ( )2 22 2 1 11 1 1 1 11131 12 442h hh h yh xh hhyxhg gA C C ggC a           = + + + + + + +           − +       ( )2 22 1 12 2 1 2 231 12 4h hh h xh hg gB C g         = + + + + + +             and DRAZ ET AL 11  ( )( )( )21 12 1 1 1 1 11 222 22 1 11 2 11312 8 2 2 2132 8 2h h h hh h xhh hyxh hg g g gF Cg gC g        +     = + + + + + +           +  + + + − + +         The optimum values of k1h and k2h obtained by minimizing (27) are, respectively, given by   ( ) ( )( )1 opt 2 opt2 2andh h h h hh h h hh hh h h h h hY A G D FB D F Gk kA B F A B F−−= =− −   Substituting the optimum values of k1h and k2h into (26) and (27) yields the minimum bias and MSE of y̅DS, given by   ( )( )2 2DS 2min12Bias 1Lh h h h h h hh hh h h hA G B D D F Gy W YA B F= + −  − −−     (28)  and   ( )( )2 22 2DS 2min12MSE 1Lh h h h h h hh hh h h hA G B D D F Gy W YA B F= + −  −−     (29) Comparison of Estimators Now, compare the proposed class of estimators y̅DS with existing estimators y̅st, y̅Rs, y̅lrs, y̅h1s, y̅h2s, y̅m1s, and y̅m2s.  Condition (i) By (2) and (29), V(y̅st) > MSE(y̅DS)min if   ( )2 22 2 22121 0Lh h h h h h hh h h yhh h h hB D A G D F GW Y CA B F= + − − + −       ESTIMATION OF FINITE POPULATION MEAN 12 Condition (ii) By (5) and (29), MSE(y̅Rs) > MSE(y̅DS)min if  ( )( )2 22 2 2 22122 1 0Lh h h h h h hh h h yh xh yxhh h h hB D A G D F GW Y C C CA B F= + − + − − + −      Condition (iii) By (8) and (29), MSE(y̅lrs)min > MSE(y̅DS)min if   ( )( )2 22 2 2 22121 1 0Lh h h h h h hh h h yh yxhh h h hB D A G D F GW Y CA B F = + − − − + −      Condition (iv) By (15) and (29), MSE(y̅h1s) > MSE(y̅DS)min if   ( )( )2 22 2 2 2 21 1 2122 1 0Lh h h h h h hh h h yh h xh h yxhh h h hB D A G D F GW Y C t C t CA B F= + − + − − + −      Condition (v) By (16) and (29), MSE(y̅h2s) > MSE(y̅DS)min if   ( )( )2 22 2 2 2 22 2 2122 1 0Lh h h h h h hh h h yh h xh h yxhh h h hB D A G D F GW Y C t C t CA B F= + − + − − + −      Condition (vi) By (22) and (29), MSE(y̅m1s) > MSE(y̅DS)min if   ( )( )2 22 2 2 2 21 1 2122 1 0Lh h h h h h hh h h yh h xh h yxhh h h hB D A G D F GW Y C C C C CA B F= + − + − − + −      Condition (vii) By (23) and (29), MSE(y̅m2s) > MSE(y̅DS)min if   ( )( )2 22 2 2 2 22 2 2122 1 0Lh h h h h h hh h h yh h xh h yxhh h h hB D A G D F GW Y C C C C CA B F= + − + − − + −     DRAZ ET AL 13 Numerical Examples Two data sets are used to obtain the percent relative efficiency (PRE) of the different estimators: Data Set 1. (Bureau of Statistics, 2013, p. 226)  Y: Employment level in 2012 by divisions. X: Number of registered factories in 2012 by divisions.  The data are divided into two groups:  Group 1: Gujranwala, Lahore, Rawalpindi, and Sargodha divisions. Group 2: Bahawalpur, D.G Khan, Faisalabad, Multan, and Sahiwal divisions.  The summary statistics are given below:  N1 = 18, N2 = 18, n1 = 8, n2 = 8, Y̅1 = 85572.11, Y̅2 = 19293.61, X̅1 = 414.5556, X̅2 = 257, XM1 = 2055, XM2 = 1674, Xm1 = 24, Xm2 = 52, Sy1 = 248216, Sy2 = 37979.33, Sx1 = 521.6751, Sx2 = 365.6955, ρyx1 = 0.3473, ρyx2 = 0.9796, Cx1 = 1.2584, Cx2 = 1.423, Cy1 = 2.9007, Cy2 = 1.9685  Data Set 2. (Bureau of Statistics, 2013, p. 135)  Y: Total number of students enrolled in 2012 by divisions. X: Total number of government primary and secondary schools for boys and girls in 2012 by divisions.  The data are divided into two groups:  Group 1: Gujranwala, Lahore, Rawalpindi, and Sargodha divisions. Group 2: Bahawalpur, D.G Khan, Faisalabad, Multan, and Sahiwal divisions.  The summary statistics are given below:  N1 = 18, N2 = 18, n1 = 8, n2 = 8, Y̅1 = 162979.3, Y̅2 = 134458, X̅1 = 962.0556, X̅2 = 1146.722, Sy1 = 255887.7, Sy2 = 50235.82, Sx1 = 307.9531, Sx2 = 469.9311, XM1 = 1530, XM2 = 2370, Xm1 = 388, Xm2 = 582, ρyx1 = 0.1447, ρyx2 = 0.787, Cx1 = 0.3202, Cx2 = 0.4098, Cy1 = 1.5701, Cy2 = 0.3736 ESTIMATION OF FINITE POPULATION MEAN 14 Table 2. Percent relative efficiency of different estimators with respect to y̅st  Estimator Data set 1 Data set 2 y̅st 100.000 100.000 y̅Rs 115.073 103.756 y̅lrs 116.252 104.509 y̅h1s 115.298 104.495 y̅h2s 105.080 103.318 y̅m1s 115.883 103.776 y̅m2s 115.880 103.778 y̅DS1 686.109 2142.170 y̅DS2 131154.840 49578.150 y̅DS3 1106.640 1724.010 y̅DS4 75600.490 29063.170 y̅DS5 893.565 3726.730 y̅DS6 2145.380 2142.520 y̅DS7 770.943 1235.560 y̅DS8 782.772 2920.860   We use the following expression for efficiency comparisons:   ( )( )stV100MSE kyPREy=     where k is one of Rs, lrs, h1s, h2s, m1s, m2s, or DSi (i = 1, 2,…, 8). The percent relative efficiencies are summarized for two data sets in Table 2. Conclusion An improved class of estimators was proposed in estimating the finite population mean using known minimum and maximum values of the auxiliary variable. Shown in Table 2 are the percent relative efficiencies of all estimators over the usual mean per unit estimator y̅st in stratified random sampling. It was observed that the performance of the suggested class of estimators y̅DSi (i = 1, 2,…, 8) is better as compared to all other considered estimators. Among the suggested class of estimators, y̅DS2 is preferable because of least MSE. References Bureau of Statistics. (2013). Punjab development statistics. Lahore, Pakistan: Bureau of Statistics, Government of the Punjab. Retrieved from http://www.bos.gop.pk/system/files/Dev-2013.pdf DRAZ ET AL 15 Cekim, H. O., & Cingi, H. (2016). Some estimator types for population mean using linear transformation with the help of the minimum and maximum values of the auxiliary variable. Hacettepe Journal of Mathematics and Statistics, 46(4). doi: 10.15672/hjms.201510114186 Khan, M. (2015). Improvement in estimating the finite population mean under maximum and minimum values in double sampling scheme. Journal of Statistics Applications & Probability Letters, 2(2), 115-121. Retrieved from http://www.naturalspublishing.com/ContIss.asp?IssID=262 Khan, M., & Shabbir, J. (2013). Some improved ratio, product, and regression estimators of finite population mean when using minimum and maximum values. The Scientific World Journal, 2013, 431868. doi: 10.1155/2013/431868 Koyuncu, N. (2012). Efficient estimators of population mean using auxiliary attributes. Applied Mathematics and Computation, 218(22), 10900-10905. doi: 10.1016/j.amc.2012.04.050 Mohanty, S., & Sahoo, J. (1995). A note on improving the ratio method of estimation through linear transformation using certain known population parameters. Sankhyā: The Indian Journal of Statistics, Series B, 57(1), 93-102. Available from https://www.jstor.org/stable/25052879 Walia, G. S., Kaur, H., & Sharma, M. (2015). Ratio type estimator of population mean through efficient linear transformation. American Journal of Mathematics and Statistics, 5(3), 144-149. Retrieved from http://article.sapub.org/10.5923.j.ajms.20150503.06.html 

Estimation of finite population mean by using minimum and maximum values in stratified random sampling

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Estimation of finite population mean by using minimum and maximum values in stratified random sampling

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