# Dual Use of Auxiliary Information for Estimating the Finite Population Mean under the Stratified Random Sampling Scheme

Authors: Sohaib Ahmad; Sardar Hussain; Muhammad Aamir; Uzma Yasmeen; Javid Shabbir; Zubair Ahmad
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we proposed an improved family of estimators for finite population mean under stratified random sampling, which needed a helping variable on the sample mean and rank of the auxiliary variable. The expression of the bias and mean square error of the proposed and existing estimators are computed up to the first-order approximation. The estimators proposed in different situations were investigated and provided a minimum mean square error relative to all other estimators considered. Four actual data sets and simulation studies are carried out to observe the performance of the estimators. For simulation study, R software is used. The mean square errors of all four data sets are minimum and percent relative efficiencies are more than a hundred percent higher than the other existing estimators, which indicated the importance of the newly proposed family of estimators. From the simulation study, it is concluded that the suggested family of estimators achieved better results. We demonstrate theoretically and numerically that the proposed estimator produces efficient results compared to all other contend estimators in entire situations. Overall, we conclude that the performance of the family of suggested estimators is better than all existing estimators.


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## Body

## 1. Introduction

In the literature of survey sampling, the use of auxiliary variables was discussed by many researchers to improve the efficiency of their developed estimators, for estimating some usual parameters, such as mean, median, variance, and standard deviation. In such situations, traditional ratio, product, and regression estimators give efficient outcomes of the unspecified parameters. There is a situation where the strength of linear association between the study and auxiliary variable is positive; then the rank of the auxiliary variable is also positively correlated with the study variable. In this article, we developed a family of the estimator for estimating mean that utilizes supplementary variable, not only in the shape of auxiliary variable but also on the rank of auxiliary variable.If the population of interest is homogeneous, then simple random sampling performs reasonably well. However, when the population of interest is heterogeneous, in such a situation it is advisable to use stratified random sampling instead of simple random sampling. In stratified random sampling, we split the whole aggregate into several nonoverlapping groups or subgroups called strata. These groups are homogeneous entirely and a sample is drawn independently from each stratum separately. To obtain the maximum benefit from stratification, the values of theNh must be known. When the strata have been determined, a sample is drawn from each stratum and the drawings are being made independently. If a simple random sample is taken from each stratum, then the entire procedure is described as stratified random sampling. Different researchers used different sample allocation techniques to distribute the sample in different strata. The use of a separate ratio estimate in each stratum is more precise if the sample in each stratum is large enough, so in this article, we use the proportional allocation method. More efforts have been pointed out about the population mean under stratified random sampling. Stratification improves efficiency when the variance between strata is much larger than the variances within strata as stated by Zaman and Kadilar [1]. More studies regarding population mean using auxiliary variables are Hussain et al. [2], Aladag and Cingi [3], Grover and Kaur [4], Shabbir and Gupta [5], Rao [6], Singh et al. [7], Kadilar and Cingi [8], Khalid [9, 10], Zaman and Bulut [11], Al-Marzouki [12], and Aamir et al. [13, 14].In this paper, we propose a new family of estimators for estimating the mean using the information on the mean and ranks of the auxiliary variable based on stratified random sampling. The remaining document is set as follows.In Section2, notations and symbols are described. The literature review and proposed estimator are given in Sections 3 and 4. In Sections 5 and 6, theoretical and numerical identification are made, while Section 7 consists of the simulation study. In Section 8, discussion of the numerical results is provided. Concluding remarks of the article are given in Section 9.


## 2. Notations and Symbols

SupposeT=1,2,…,N is the population of N definite units, which is partitions into L strata, where the size of hth stratum is Nh, for h=1,2,…,L such that ∑h=1LNh=N. Let Y and X be the study and auxiliary variables which take values yh and xh, respectively, where i=1,2,…,Nh and h=1,2,…,L; for estimating mean, suppose a sample of size nh is taken from the hth stratum using SRSWOR. ∑h=1Lnh=n, where n is the sample size. Consider Y the study variable, X and Z the auxiliary and rank of the auxiliary variable, Y˜h=∑i=1NhYih/Nh, X˜h=∑i=1NhXih/Nh, and Z˜h=∑i=1NhZih/Nh the population mean of Y, X, and Z for the hth stratum,Y˜st=Y˜=∑h=1LWhX˜h, X˜st=X˜=∑h=1LWhX˜h, and Z˜st=Z˜=∑h=1LWhZ˜h the population mean of Y, X, and Z based on stratified random sampling, Y˜^h=∑i=1nhYih/nh, X˜^h=∑i=1nhXih/nh, and Z˜^h=∑i=1nhZih/nh the sample mean of Y, X, and Z for the hth stratum, Y˜^st=Y˜^=∑h=1LWhY˜^h ,X˜^st=X˜^=∑h=1LWhX˜^h, and Z˜^st=Z˜^=∑h=1LWhZ˜^h the sample mean of Y, X, and Z based on stratified random sampling, Syh2=∑i=1NhYih−Y˜h2/Nh−1, Sxh2=∑i=1NhXih−X˜h2/Nh−1, and Szh2=∑i=1NhZih−Z˜h2/Nh−1 the population variance of Y, X, and Z for the hth stratum, Cyh=Syh/Y˜h, Cxh=Sxh/X˜h, and Czh=Szh/Z˜h the population coefficient of variation of Y, X, and Z for the hth stratum, Syxh=∑i=1NhYih−Y˜hXih−X˜h/Nh−1, Syzh=∑i=1NhYih−Y˜hZih−Z˜hh/Nh−1, and Sxzh=∑i=1NhXih−XhZih−Z˜h/Nh−1 the covariance between (Y, X), (Y, Z), and (X, Z) for the hth stratum, Ryxh=Syxh/SyhSxh, Ryzh=Syzh/SyhSzh, and Rxzh=Sxzh/SxhSzh the population correlation between (Y, X), (Y, Z), and (X, Z) for the hth stratum, Ryx=∑h=1LWh2λhRyxhSyhSxh/∑h=1LWh2λhSyh2∑h=1LWh2λhSxh2 the population correlation coefficient between Y and X based on stratified random sampling, Ryz=∑h=1LWh2λhRyzhSyhSzh/∑h=1LWh2λhSyh2∑h=1LWh2λhSzh2 the correlation coefficient between Y and Z based on stratified random sampling, Rxz=∑h=1LWh2λhRxzhS2hSxh/∑h=1LWh2λhSxh2∑h=1LWh2λhSzh2 the correlation coefficient between X and Z based on stratified random sampling, and Ry.xz2=Ryx2+Ryz2−2RyxRyzRxz/1−Rxz2 the strength of the linear relationship of Y on X and Z.To obtain the properties of the existing and proposed estimators ofY˜, we consider the following relative error terms based on stratified random sampling.Lete1=Y˜^−Y˜/Y˜ , e2=X˜^−X˜/X˜, and e3=Z˜^−Z˜/Z˜, such that Eei=0 for i=1,2,3.(1)Ee12=∑h=1LWh2λh2Cyh2=V200,Ee22=∑h=1LWh2λh2Cxh2=V020,Ee32=∑h=1LWh2λh2Czh2=V002,Ee1e2=∑h=1LWh2λh2RyxhCyhCxh=V110,Ee1e3=∑h=1LWh2λh2RyzhCyhCxh=V101,Ee2e3=∑h=1LWh2λh2RxzhCxhCzh=V011.


## 3. Existing Estimators

In this portion, we consider some estimators of population mean based on stratified sampling schemes that are available in the literature. The properties of these existing estimators are obtained up to the first degree of approximation. Different estimators with their error terms and mean square errors are as follows:(1)
The usual estimator ofY˜ is given by(2)Y˜^SRSst=1n∑i=1nYi=Y˜^.The variance ofY˜^SRSst is given by(3)VarY˜^SRSst=Y˜2V200.(2)
The existing ratio estimator ofY˜ is given by Cochran [15].(4)Y˜^Rst=Y˜^X˜X˜^st.The properties ofY˜^Rst are given by(5)BiasY˜^Rst≅Y˜V020−V110,MSEY˜^Rst≅Y˜2V200+V020−2V110.(3)
The existing product estimator ofY˜ is given by Murthy [16].(6)Y˜^Pst=Y˜^X˜^stX˜.The properties ofY˜^Pst given as(7)BY˜^Pst=Y˜V110,MSEY˜^Pst≅Y˜2V200+V020+2V110.(4)
The existing regression estimator ofY˜ is given by(8)Y˜^Rgst=Y˜^st+QX˜−X˜^st.The minimum variance ofX˜^Regst at the optimal value Q is given by(9)VarminY˜Regst=Y˜2V200V020−V1102V020.Equation (9) will become(10)VarminY˜^Regst=Y˜2V2001−Ryx2.(5)
The existing difference-type estimator ofY˜ is given by(11)Y˜^R,Dst=Q1Y˜^st+Q2X˜−X˜^st,whereQ1 and Q2 are unknown constants. The bias and MSE of Y˜^R,Dst are given as(12)BY˜^R,Dst=Y˜Q1−1,MSEY˜^R,Dst≅Y˜2−2Q1Y˜2+Q12Y˜2+Q12Y˜2V200−2Q1Q2Y˜X˜V110+Q22X˜2V020.The optimal values ofQ1 and Q2 are given as(13)Q1opt=V020V020V200−V1102+V020,Q2opt=Y˜V110X˜V200V020−V1102+V020.The minimum MSE ofY˜^R,Dst at the optimal values of Q1, Q2 is written as(14)MSEminY˜^R,Dst=Y˜2V200V020−V1102V200V020−V1102+V020.Equation (14) can also be written as(15)MSEminY˜^R,Dst=Y˜2V2001−Ryx21+V2001−Ryx2.(6)
Exponential estimator [13] is given by(16)Y˜^Sst=Y˜^stexpaX˜−X˜^staX˜+X˜^st+2b.The properties ofY˜^Sst is given by(17)BY˜^Sst≅Y˜38θ2V020−12θV110,MSEY˜^Sst≅Y˜244V200+θ2V020−4θV110,where θ=aX˜/aX˜+b.(7)
Grover and Kaur [17] are given by(18)Y˜^gr,krst=Q3Y˜^st+Q4X˜−X˜^stexpaX˜−X˜^staX˜+X˜^st+2b.The properties ofY˜^gr,krst are given by(19)BY˜^gr,krst≅Y˜Q3−1+38θ2Q3Y˜+12θQ4X˜V020−12θY˜V110,MSEY˜^gr,krst≅Q42X˜2V020+Q32Y˜2V200+2θQ3Q4Y˜X˜V020−2Q3Q4Y˜X˜V110+Y˜2−2Q3Y˜2+θQ32Y˜2+Q3Y˜2V110−θQ4Y˜X˜V020−2θQ32Y˜2V110−34θ2Q3Y˜2V020+θ2Q32Y˜2V020.From (19), the optimal values are given by(20)Q3opt=V020θ2V020−88−V200V020+V1102−V020,Q4opt=Y˜θ3V0202−θ2V020V110+4θV200V020−4θV1102−4θV020+8V1108X˜V200V020−V1102+V020.The minimum MSE ofY˜^gr,krst at the optimal values is given by(21)MSEminY˜^gr,krst≅Y˜26464−16θ2V020−V020−8+θ2V0202V0201+V200−V11002,or(22)MSEminY˜^gr,krst≅VarminY˜^Regst−Y˜2θ2V0202−8V1102+8V020V200264V02021+V2001−Ryx2,and Y˜^gr,krst is more efficient as compared to Y˜^Regst.


## 4. Proposed Estimator

The use of auxiliary variables may improve the efficiency of an estimator in the design or estimation stages. When there is a correlation between the study and the auxiliary variable, the rank of the auxiliary variable is correlated with the study variables as well. Therefore, the rank of the auxiliary variable can be treated as a new auxiliary variable, and it is useful to increase the effectiveness of an estimator. By taking motivation from Hussain et al. [2], we developed a new family of estimators of Y˜ which needed a helping variable on the sample mean and rank of the auxiliary variable. Moreover, some members of our proposed and existing estimates are also presented in Table 1.(23)Y˜^Prst=Q5Y˜^st+Q6X˜−X˜^stX˜+Q7Z˜−Z˜^stZ˜expaX˜−X˜^staX˜+X˜^st+2b,where Q5, Q6, and Q7 are unknown values.Table 1
Members of our proposed and existing estimator.
abY˜^SstY˜^gr,krstY˜^Prst1CxstY˜^Sst1Y˜^gr,krst1Y˜^Prst11β2stY˜^Sst2Y˜^gr,krst2Y˜^Prst2β2stCxstY˜^Sst3Y˜^gr,krst3Y˜^Prst3Cxstβ2stY˜^Sst4Y˜^gr,krst4Y˜^Prst41RyxY˜^Sst5Y˜^gr,krst5Y˜^Prst5CxstRyxY˜^Sst6Y˜^gr,krst6Y˜^Prst6RyxCxstY˜^Sst7Y˜^gr,krst7Y˜^Prst7β2stRyxY˜^Sst8Y˜^gr,krst8Y˜^Prst8Ryxβ2stY˜^Sst9Y˜^gr,krst9Y˜^Prst91NX˜Y˜^Sst10Y˜^gr,krst10Y˜^Prst10Equation (23) can be expressed as(24)Y˜^Prst=Q5Y˜1+e1−Q6e2−Q7e31−12θe2+38θ2e22+⋯.After simplification (24), we have(25)Y˜^Prst−Y˜=−Y˜+Q5Y˜+Q5Y˜e1−12θQ5Y˜e2−Q6e2−Q7e3+38θ2Q5Y˜e12+12θQ6e12−12θQ5Y˜e1e2+12θQ7e2e3.The properties ofY˜^Prst are shown as(26)BY˜^Prst≅Y˜Q5−1+38θ2Q5Y˜V020+12θQ6V020−12θQ5Y˜V110+12θV011MSEY˜^Prst≅Y˜2Q5−12+Q52Y˜2V200+Q6V020+Q7V002+θ2Q52Y˜2V020−θQ6Y˜V020+2θY˜V020−34θ2Q5Y˜2V020+θQ5Y˜2V110−2θQ52Y˜2V110−2Q5Q6Y˜V110−2Q5Q7Y˜V101−θQ7Y˜V011+2θQ5Q7Y˜V011−2Q6Q7V011.The optimal values ofQ5, Q6, and Q7, determined by reducing (26), are given by(27)Q5opt=8−θ2V02081+V2001−Ry.xz2,Q6opt=Y˜θ3V0203/2Rxz2−1+V2001/2−8+θ2V020Ryx−RxzRyz+4θV0201/2Rxz2−1−1+V2001−Ry.xz28V0201/2Rxz2−1−1+V2001−Ry.xz2,Q7opt=Y˜V2001/28−θ2V020Ryx−RxzRyz8V0201/2Rxz2−1−1+V2001−Ry.xz2.The simplified least mean square error ofY˜^Prst at optimal values of Q5, Q6, and Q7 is as follows:(28)MSEminY˜^Prst≅Y˜264V2001−Ry.xz2−θ4V0202−16θ2V020V2001−Ry.xz2641+V2001−Ry.xz2,where R1.232=V1102V002+V1012V020−2V101V110V011/V200V020V002−V0112.Here, (28) may be written as(29)MSEminY˜^Prst≅VarminY˜^Regst−T1−T2,where(30)T1=Y˜2θ2V0202−8V1102+8V020V200264V02021+V2001−Ryx2,T2=Y˜2θ2V020−82V020V101−V011V110264V0202V0021−Rxz21+V2001−Ryx21+V2001−R1.232.


## 5. Efficiency Comparisons

In the following portion, we made an efficiency comparison of all estimators.(i)
By taking (3) and (29),(31)MSEminY˜^Prst<VarY˜^SRSstifY˜2V2002Ryx2+T1+T2>0.(ii)
By taking (5) and (29),(32)MSEminY˜^Prst<MSEY˜^Rstif1V020V020−V1102+T1+T2>0.(iii)
By taking (7) and (29),(33)MSEminY˜^Prst<MSEY˜^Pstif1V020V020+V1102+T1+T2>0.(iv)
By taking (10) and (29),(34)MSEminY˜^Prst<MSEminY˜^RegstifT1+T2>0.(v)
By taking (15) and (29),(35)MSEminY˜^Prst<MSEminY˜^R,DstifY˜2θ2V020θ2V020+16V2001−Ryx2641+1−Ryx2+T2>0.(vi)
By taking (17) and (29),(36)MSEminY˜^Prst<MSEY˜^Sstif1V020θV0202−V1102+T1+T2>0.(vii)
By taking (22) and (29),(37)MSEminY˜^Prst<MSEminY˜^gr,krstifT2>0.


## 6. Numerical Investigation

In this portion, the mathematical outcome is illustrated to check the efficiency of all estimators. Four data sets are contemplated. The finding outcomes of these data sets are listed in Tables2–5. The percent relative efficiency of the estimator Y˜^i w.r.t. Y˜^SRSst is as follows:(38)PREY˜^i,Y˜^SRSst=VarY˜^SRSstMSEminY˜^i×100,where i=Rst,Pst,…,Prst.Table 2
Data description of population 1 (scenario I).
hNhnhWhλhY˜hX˜hZ˜hSyhSxhSzhRyxhRyzhRxzh1127310.13750.02447042080564883.83486.756.800.93660.82390.78342117210.12670.0390413921259644.925180.773.920.99560.65840.65173103290.11150.02487414309521033.4627549.79.870.99370.63370.62374170380.18410.0204425947986810.588218.9349.210.98340.63600.64425205220.22210.04062675570103403.658497.7759.320.98930.65950.66556201390.21770.020739412998101711.723094.1458.160.96510.58630.6162Table 3
Data description of population 1 (scenario II).
hNhnhWhλhY˜hX˜hZ˜hSyhSxhSzhRyxhRyzhRxzh1127310.13750.024470449864883.8355.5836.800.93660.82390.78342117210.12670.039141331859644.9265.4533.920.99560.65840.65173103290.11150.0248574431521033.4612.9529.870.99370.63370.62374170380.18410.020442531186810.58458.0249.220.98340.63600.64425205220.22210.0406267227103410.6560.8559.320.98930.65950.66556201390.21770.0207394314101711.7297.0558.160.96510.58630.6162Table 4
Data description of population 2 (scenario I).
HNhnhWhλhY˜hX˜hZ˜hSyhSxhSzhRyxhRyzhRxzh110690.12410.1017153724376546425.0849189.0830.740.81560.33490.59302106170.12410.04942213274225411551.5357460.6130.740.85590.28160.6031394380.11000.01579384724104829907.48160757.3127.280.90110.46370.58734171670.20020.00905588743658728643.42285603.1249.510.98580.29810.3654520470.23890.4657967264421032389.7745402.7845.410.71300.45470.6206617320.20260.4942404984487945.7418793.9650.080.89350.54350.6262Table 5
Data description of population 2 (scenario II).
hNhnhWhλhY˜hX˜hZ˜hSyhSxhSzhRyxhRyzhRxzh110690.12410.1017153724712546425.0849134.7630.740.81560.33460.59562106170.12410.04942213268405411551.5353978.7130.740.83590.28140.6246394380.11000.01579384727224829907.48161109.5027.270.89710.46260.58934171670.20020.00905588731918728643.42262495.6149.500.98140.29790.3885520470.23890.1379967268341032389.7745174.2659.030.71070.45410.6317617320.20260.4942404990387945.7418977.2850.080.86970.53660.6283The percent relative efficiency of the finite population means, measured from four data sets, is listed in Tables6–9.Population 1 (source: Koyuncu and Kadilar [18]): Y is the number of instructors and X is the number of the trainees in 2007 for 923 districts in six regions.Population 2 (source: Kadilar and Cingi [19]): Y is yield of apples in 1999 and X is yield of apples in 1998.Table 6
MSEs using population 1 (scenario I).
EstimatorMSEEstimatorMSEEstimatorMSEEstimatorMSEY˜^SRSst2229.266Y˜^Sst15096.365Y˜^gr,krst1192.9490Y˜^Prst1185.1848Y˜^Rst9205.298Y˜^Sst2604.1097Y˜^gr,krst2192.9539Y˜^Prst2185.1897Y˜^Pst216.4183Y˜^Sst3602.4522Y˜^gr,krst3192.9485Y˜^Prst3185.1844Y˜^Regst194.2832Y˜^Sst4603.4212Y˜^gr,krst4192.9516Y˜^Prst4185.1875Y˜^R,Dst194.0853Y˜^Sst5602.5302Y˜^gr,krst5192.9487Y˜^Prst5185.1846Y˜^Sst6602.4947Y˜^gr,krst6192.9486Y˜^Prst6185.1845Y˜^Sst7602.5978Y˜^gr,krst7192.9490Y˜^Prst7185.1849Y˜^Sst8602.4488Y˜^gr,krst8192.9485Y˜^Prst8185.1844Y˜^Sst9604.1488Y˜^gr,krst9192.9540Y˜^Prst9185.1898Y˜^Sst102226.835Y˜^gr,krst10194.0853Y˜^Prst10186.2958Table 7
MSEs using population 1 (scenario II).
EstimatorMSEEstimatorMSEEstimatorMSEEstimatorMSEY˜^SRSst2229.266Y˜^Sst14240.170Y˜^gr,krst1101.1275Y˜^Prst179.35476Y˜^Rst6936.636Y˜^Sst2914.5469Y˜^gr,krst2101.1566Y˜^Prst279.38143Y˜^Pst193.2885Y˜^Sst3877.6616Y˜^gr,krst3101.1242Y˜^Prst379.35178Y˜^Regst101.5021Y˜^Sst4907.0444Y˜^gr,krst4101.1503Y˜^Prst479.37566Y˜^R,Dst101.4481Y˜^Sst5880.3341Y˜^gr,krst5101.1267Y˜^Prst579.35404Y˜^Sst6879.7216Y˜^gr,krst6101.1261Y˜^Prst679.35352Y˜^Sst7881.2793Y˜^gr,krst7101.1276Y˜^Prst779.35483Y˜^Sst8877.5927Y˜^gr,krst8101.1242Y˜^Prst879.35172Y˜^Sst9915.3844Y˜^gr,krst9101.1573Y˜^Prst979.38207Y˜^Sst102227.442Y˜^gr,krst10101.4481Y˜^Prst1079.63742Table 8
MSEs using population 2 (scenario I).
EstimatorMSEEstimatorMSEEstimatorMSEEstimatorMSEY˜^SRSst697789.6Y˜^Sst11225952Y˜^gr,krst1214502.1Y˜^Prst1197279.1Y˜^Rst1949400Y˜^Sst2365071.7Y˜^gr,krst2214508.7Y˜^Prst2197285.3Y˜^Pst227015.9Y˜^Sst3364798.4Y˜^gr,krst3214501.7Y˜^Prst3197278.7Y˜^Regst222881.3Y˜^Sst4364915.7Y˜^gr,krst4214504.7Y˜^Prst4197281.5Y˜^R,Dst217241.8Y˜^Sst5364803.5Y˜^gr,krst5214501.8Y˜^Prst5197278.9Y˜^Sst6364800.4Y˜^gr,krst6214501.8Y˜^Prst6197278.8Y˜^Sst7364815.1Y˜^gr,krst7214502.1Y˜^Prst7197279.1Y˜^Sst8364798.2Y˜^gr,krst8214501.7Y˜^Prst8197278.7Y˜^Sst9365117.8Y˜^gr,krst9214509.9Y˜^Prst9197286.4Y˜^Sst10697286.1Y˜^gr,krst10217241.8Y˜^Prst10199821.3Table 9
MSEs using population 2 (scenario II).
EstimatorMSEEstimatorMSEEstimatorMSEEstimatorMSEY˜^SRSst697789.6Y˜^Sst11225952Y˜^gr,krst1228480.8Y˜^Prst1207876.6Y˜^Rst1878241Y˜^Sst2365071.7Y˜^gr,krst2228486.8Y˜^Prst2207882.2Y˜^Pst243316.9Y˜^Sst3364798.4Y˜^gr,krst3228480.4Y˜^Prst3207876.3Y˜^Regst237552.8Y˜^Sst4364915.7Y˜^gr,krst4228483.3Y˜^Prst4207878.9Y˜^R,Dst231157.0Y˜^Sst5364803.5Y˜^gr,krst5228480.6Y˜^Prst5207876.4Y˜^Sst6364800.4Y˜^gr,krst6228480.5Y˜^Prst6207876.3Y˜^Sst7364815.1Y˜^gr,krst7228480.8Y˜^Prst7207876.7Y˜^Sst8364798.2Y˜^gr,krst8228480.4Y˜^Prst8207876.3Y˜^Sst9365117.8Y˜^gr,krst9228488.0Y˜^Prst9207883.3Y˜^Sst10697286.1Y˜^gr,krst10231157.0Y˜^Prst10210333.4


## 7. Simulation Study

A simulation study is performed to determine the efficiency of the estimators suggested for stratified sampling technique with the help of information of single auxiliary variableX and adapted rank set sampling procedure on auxiliary variable X and then made variable Z. In this section, the performance of the suggested exponential cum ratio type family of estimators is appraised for the artificially generated population of different stratum and sample sizes. The population elements are randomly generated from N1∼2,6 and N1∼3,9 compared to that from the values of x1 and x2 generated y1 and y2. Applying the procedure of ranked set sampling, we obtained Z variable. The comparison is offered in the form of percentage relative efficiency (PRE). The relative efficiency in Table 10 is computed using the following formula:(39)PREY˜^i,Y˜^SRSst=VarY˜^SRSstMSEminY˜^i×100,where i=Rst,Pst,…,Prst.Table 10
Simulation result for the percent relative efficiencies of the suggested family of exponential cum ratio type estimators w.r.t. existing estimators for different stratum.
EstimatorPREEstimatorPREEstimatorPREEstimatorPREY˜^SRSst100.00Y˜^Sst1145.79Y˜^gr,krst1186.6Y˜^Prst11518.9Y˜^Rst156.74Y˜^Sst2100.21Y˜^gr,krst2208.9Y˜^Prst21718.9Y˜^Pst101.98Y˜^Sst3170.96Y˜^gr,krst3754.9Y˜^Prst31280.78Y˜^Regst145.62Y˜^Sst4149.49Y˜^gr,krst4416.10Y˜^Prst41281.55Y˜^R,Dst137.62Y˜^Sst5110.21Y˜^gr,krst5209.19Y˜^Prst51291.51Y˜^Sst6100.11Y˜^gr,krst6210.19Y˜^Prst61254.59Y˜^Sst7179.00Y˜^gr,krst7861.9Y˜^Prst71921.51Y˜^Sst8100.12Y˜^gr,krst8254.9Y˜^Prst81890.59Y˜^Sst9162.35Y˜^gr,krst9233.78Y˜^Prst91489.01Y˜^Sst10172.54Y˜^gr,krst10243.58Y˜^Prst101220.45In this study, consider the generated population for summarizing the simulation procedures. The process is used to find out the efficiency of the family of suggested estimators over all the existing estimators.From the described generated population of the normal population, the sample size is considered asn1=25 and n2=25. We take r=10 and m=50 for the ranks of the auxiliary variable x. The procedure has been repeated 10000 times and the whole aggregate is split into two strata, where N=1000 and N1=500, N2=500 evaluate several available families of existing estimators under stratified random sampling technique. Using the sample, simulate 10,000 times estimators values of existing estimators class Y˜^Sst1, Y˜^Sst2, Y˜^Sst3, Y˜^Sst4, Y˜^Sst5, Y˜^Sst6, Y˜^Sst7, Y˜^Sst8, Y˜^Sst9, Y˜^Sst10, and Y˜^gr,krst1, Y˜^gr,krst2, Y˜^gr,krst3, Y˜^gr,krst4, Y˜^gr,krst5, Y˜^gr,krst6, Y˜^gr,krst7, Y˜^gr,krst8, Y˜^gr,krst9, Y˜^gr,krst10, and Y˜^Sst1, Y˜^Sst2, Y˜^Sst3, Y˜^Sst4, Y˜^Sst5, Y˜^Sst6, Y˜^Sst7, Y˜^Sst8, Y˜^Sst9, Y˜^Sst10, separately are obtained for the formula described in (39), respectively. This simulation study is evaluating the percent relative efficiencies of the suggested exponential cum ratio type family of estimators and available estimators for sample sizes in two strata. From the critical comparison, it is concluded that the suggested family of estimators perform better result. Overall, we can conclude the performance of the family of suggested estimators is better than all existing estimators.


## 8. Discussion

As mentioned above, we used two real data sets and a simulation study to determine the efficiency of the estimators suggested for stratified sampling technique with the help of information of single auxiliary variableX and rank of the auxiliary variable. We also considered different sample sizes from the populations for scenario I and scenario II. The proposed family of estimators and the adapted stratified estimators were compared to each other with respect to their mean square error and percentage relative efficiency. In Tables 2–5, we present the data description. The results of mean square error for the real data sets are presented in Tables 6–9. And the percentage relative efficiency is shown in Tables 11–14.Table 11
PREs of finite population mean estimators using population 1 (scenario I).
EstimatorPREEstimatorPREEstimatorPREEstimatorPREY˜^SRSst100.00Y˜^Sst143.74Y˜^gr,krst11155.37Y˜^Prst11203.81Y˜^Rst24.22Y˜^Sst2369.02Y˜^gr,krst21155.34Y˜^Prst21203.77Y˜^Pst1030.07Y˜^Sst3370.03Y˜^gr,krst31155.37Y˜^Prst31203.81Y˜^Regst1147.43Y˜^Sst4369.44Y˜^gr,krst41155.35Y˜^Prst41203.79Y˜^R,Dst1148.60Y˜^Sst5369.98Y˜^gr,krst51155.37Y˜^Prst51203.81Y˜^Sst6370.01Y˜^gr,krst61155.37Y˜^Prst61203.81Y˜^Sst7369.94Y˜^gr,krst71155.37Y˜^Prst71203.81Y˜^Sst8370.03Y˜^gr,krst81155.37Y˜^Prst81203.81Y˜^Sst9368.99Y˜^gr,krst91155.34Y˜^Prst91203.77Y˜^Sst10100.11Y˜^gr,krst101148.60Y˜^Prst101196.63Table 12
PREs of finite population mean estimators using population 1 (scenario II).
EstimatorPREEstimatorPREEstimatorPREEstimatorPREY˜^SRSst100.00Y˜^Sst152.57Y˜^gr,krst12204.41Y˜^Prst12809.24Y˜^Rst32.14Y˜^Sst2243.76Y˜^gr,krst22203.78Y˜^Prst22808.30Y˜^Pst1153.34Y˜^Sst3254.00Y˜^gr,krst32204.48Y˜^Prst32809.35Y˜^Regst2196.27Y˜^Sst4245.77Y˜^gr,krst42203.92Y˜^Prst42808.50Y˜^R,Dst2197.45Y˜^Sst5253.23Y˜^gr,krst52204.43Y˜^Prst52809.27Y˜^Sst6253.41Y˜^gr,krst62204.44Y˜^Prst62809.28Y˜^Sst7252.96Y˜^gr,krst72204.41Y˜^Prst72809.24Y˜^Sst8254.02Y˜^gr,krst82204.48Y˜^Prst82809.35Y˜^Sst9243.53Y˜^gr,krst92203.76Y˜^Prst92808.27Y˜^Sst10100.08Y˜^gr,krst102197.45Y˜^Prst102799.27Table 13
PREs of finite population mean estimators using population 2 (scenario I).
EstimatorPREEstimatorPREEstimatorPREEstimatorPREY˜^SRSst100.00Y˜^Sst156.92Y˜^gr,krst1325.31Y˜^Prst1353.71Y˜^Rst35.80Y˜^Sst2191.14Y˜^gr,krst2325.30Y˜^Prst2353.70Y˜^Pst307.37Y˜^Sst3191.28Y˜^gr,krst3325.31Y˜^Prst3353.71Y˜^Regst313.08Y˜^Sst4191.22Y˜^gr,krst4325.30Y˜^Prst4353.70Y˜^R,Dst321.20Y˜^Sst5191.28Y˜^gr,krst5325.31Y˜^Prst5353.71Y˜^Sst6191.28Y˜^gr,krst6325.31Y˜^Prst6353.71Y˜^Sst7191.27Y˜^gr,krst7325.31Y˜^Prst7353.71Y˜^Sst8191.28Y˜^gr,krst8325.31Y˜^Prst8353.71Y˜^Sst9191.11Y˜^gr,krst9325.29Y˜^Prst9353.69Y˜^Sst10100.07Y˜^gr,krst10321.20Y˜^Prst10349.21Table 14
PREs of finite population mean estimators using population 2 (scenario II).
EstimatorPREEstimatorPREEstimatorPREEstimatorPREY˜^SRSst100.00Y˜^Sst158.28Y˜^gr,krst1305.40Y˜^Prst1335.67Y˜^Rst37.15Y˜^Sst2183.60Y˜^gr,krst2305.40Y˜^Prst2335.67Y˜^Pst286.78Y˜^Sst3183.72Y˜^gr,krst3305.40Y˜^Prst3335.68Y˜^Regst293.74Y˜^Sst4183.67Y˜^gr,krst4305.40Y˜^Prst4335.67Y˜^R,Dst301.87Y˜^Sst5183.72Y˜^gr,krst5305.40Y˜^Prst5335.68Y˜^Sst6183.72Y˜^gr,krst6305.40Y˜^Prst6335.68Y˜^Sst7183.71Y˜^gr,krst7305.40Y˜^Prst7335.67Y˜^Sst8183.72Y˜^gr,krst8305.40Y˜^Prst8335.68Y˜^Sst9183.58Y˜^gr,krst9305.39Y˜^Prst9335.66Y˜^Sst10100.07Y˜^gr,krst10301.87Y˜^Prst10331.75From the numerical findings, which are put in Tables6–9 and Tables 10–14, it is observed that the MSEs and PREs of our proposed family of estimators change with the choices of a and b. It is also observed that the proposed family of estimators are more precise than the existing estimators on both real data sets and simulation study, in terms of MSEs and PREs.


## 9. Concluding Remarks

In this manuscript, we proposed a new family of estimators for the finite population mean under the stratified sampling scheme, which needed a helping variable on the sample mean and rank of the auxiliary variable. Based on the real data sets, it is observed that the proposed class of estimators perform better theoretically and numerically compared to their existing counterparts and should be preferable over the existing estimators available in the literature. To check the robustness and generalizability of the proposed family of estimators, a simulation study is also conducted. The results of the simulation study also support the usefulness of the proposed family of estimators. Hence, it is recommended that the new suggested family of estimators should be used for better results when estimating population mean under stratified random sampling scheme.



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