# Improved Estimation of Finite Population Variance Using Dual Supplementary Information under Stratified Random Sampling

Authors: Sohaib Ahmad; Sardar Hussain; Javid Shabbir; Erum Zahid; Muhammad Aamir; Ronald Onyango
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this article, we propose an improved estimator for finite population variance based on stratified sampling by using the auxiliary variable as well as the rank of the auxiliary variable. Expressions for the bias and the mean square error of the estimators are derived up to the first order of approximation. Four real data sets are used to measure the performances of estimators. Moreover, a simulation study is also conducted to observe the efficiency of the proposed variance estimator. The theoretical and numerical results show that the proposed estimator under stratified random sampling is more efficient as compared to the existing estimators.


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## Body

## 1. Introduction

The need for supplementary information in survey sampling has long been acknowledged as producing efficient estimators of population parameters such as the mean, median, mode, quartiles, interquartile, percentile, coefficient of variation, and proportion. A wide amendment of methods for retaining supplementary information is described in the literature on survey sampling. The ratio, product, and regression type estimators take advantage of the correlation between the study variable and the auxiliary variable. These estimators perform better when the correlation between the study variable and the auxiliary variable exists and is often used to increase the precision of estimators. When the correlation between the study variable and the supplementary information occurs, the rank of the supplementary information is also correlated with the study variable, and therefore, this rank can be used as an essential factor for increasing the accuracy of an estimator. When the population variance of the supplementary information is known in advance, researchers have developed ratio and product-type estimators for variance estimation. In trades such as agriculture, medicine, biology, and industry, where we encounter populations that are likely to be skewed, a variance estimate for finite population parameters is considered. Variations can occur in reality in our daily lives in various fields such as environmental, genetic, and economic studies. For example, an agriculturist needs a suitable understanding of the variation in weather factors, particularly from time to time or from place to place, to be able to plan on where, how and when to plant his crop. A physician needs a full understanding of variation in the degree of human blood pressure, body temperature, and pulse rate for an adequate prescription. An industrialist desires continual knowledge of the level of variation in people’s response to his product to be capable of knowing whether to reduce or increase his price or improve the quality of his product.The issue of estimating the population variance has been broadly argued by various authors. Das and Tripathi [1] considered the use of the auxiliary variable in estimating the finite population variance. Isaki [2] discussed the variance estimation using ratio and regression-type estimators. Some important references regarding variance estimation include Garcia and Cebrian [3]; Arcos et al. [4]; Shabbir and Gupta [5]; Singh and Solanki [6]; Adichwal et al. [7, 8], Ahmad et al. [9]; Shabbir and Gupta [10]; Singh and Khalid [11, 12] and Zaman and Bulut [13].In stratified random sampling, a population is divided into a number of nonoverlapping groups or subgroups called strata. These groups are completely homogeneous, and the sample is taken independently from each stratum. Stratification increases precision when the variance among the strata is much larger than the variances within the strata. Some important references under stratified random sampling are Rao and Shao [14], Kadilar and Cingi [15, 16], Singh and Vishwakarma [17], Koyuncu and Kadilar [18], Shabbir and Gupta [19], Ozel et al. [20], Sidelel et al. [21], Ahmad and Shabbir [22], Hussain et al. [23], Shehzad et al. [24], Singh et al. [25], Zaman [26], Zaman and Bulut [27], and Ahmad et al. [28].In Section2, we discuss some notations and symbols of population variance under stratified random sampling. In Section 3, we review some adopted existing estimators. A proposed estimator is given in Section 4. The efficiency comparisons are given in Section 5. In Section 6, we discuss the numerical investigation. The simulation study is given in Section 7. The discussion of the article is discussed in Section 8. The conclusion of the paper is given in Section 9.


## 2. Notations and Symbols

Consider a finite populationψ=ψ1,ψ2,…,ψN, having N units into L strata. Let yhi, xhi, and rxhi be the characteristics of the study variable (y), auxiliary variable (x), and rank of the auxiliary variable rx, respectively, in stratum h such that ∑h=1LNh=N.We draw a random sample of sizenh from a population Nh such that ∑h=1Lnh=n. Let Y¯h, X¯h, and R¯xh be the population means corresponding to the sample means y¯h, x¯h, and r¯xh, respectively, in each stratum.Fory, x, and the rank of x, we take the values of yhi, xhi, and rxhi for the ith unit of the hth stratum.Letsyh2=1/nh−1∑i=1nhyhi−y¯h2,sxh2=1/nh−1∑i=1nhxhi−x¯h2 and srxh2=1/nh−1∑i=1nhrxhi−r¯xh2 be the sample variance, corresponding to the population variances Syh2=1/Nh−1∑i=1Nhyhi−Y¯h2,Sxh2=1/Nh−1∑i=1Nhxhi−X¯h2, and Srxh2=1/Nh−1∑i=1Nhrxhi−R¯xh2.To derive the bias and mean square error, we define the following error terms:(1)ξoh=syh2−Syh2Syh2,ξ1h=sxh2−Sxh2Sxh2,ξ2h=srxh2−Srxh2Srxh2,Eξih=0,where i=0,1,2.(2)Eξoh2=ʎhƛ400h∗,Eξ1h2=ʎhƛ040h∗,Eξ2h2=ʎhƛ004h∗,Eξohξ1h=ʎhƛ220h∗,Eξ0hξ2h=ʎhƛ202h∗,Eξ1hξ2h=ʎhƛ022h∗,where(3)ƛabch=υabchυ200ha/2υ020hb/2υ002hc/2,υabch=∑i=1Nyhi−Y¯haxhi−X¯hbrxhi−R¯xhcNh−1ƛabch∗=ƛabch−1andʎh=1nh−1Nh.


## 3. Existing Estimators

In this section, we adopt some variance estimators under stratified random sampling, which are available in the literature.(i)
The traditional unbiased estimator is given by(4)Ɠ0st=∑h=1lWh2ʎhSyh2.The variance ofƓ0st is given by(5)VarƓ0st≅∑h=1lWh4ʎh3Syh4ƛ400h∗.(ii)
Isaki [2] suggested a ratio estimator which is given by(6)Ɠ1st=∑h=1lWh2ʎhsyh2Sxh2sxh2.The bias and MSE ofƓ1st are given by(7)BiasƓ1st≅∑h=1lWh2ʎh2Syh2ƛ040h∗−ƛ220h∗,and,MSEƓ1st≅∑h=1lWh4ʎh3Syh4ƛ400h∗+ƛ040h∗−2ƛ220h∗.(iii)
The usual difference-type estimator is given by(8)Ɠ2st=∑h=1lWh2ʎhsyh2+ΨhSxh2−sxh2.WhereΨh is the unknown constant.The optimum value ofΨh is given by(9)Ψh=Syh2ƛ220h∗Sxh2ƛ040h∗.The minimum variance at the optimal value ofΨh is given by(10)VarƓ2stmin=∑h=1lWh4ʎh3Syh4ƛ400h∗1−ρh2.Where(11)ρh=ƛ220h∗ƛ400h∗ƛ040h∗.(iv)
Rao [29] proposed a difference-type estimator which is given by(12)Ɠ3st=∑h=1lWh2ʎhϢ1hsyh2+Ϣ2hSxh2−sxh2,where,Ϣ1h and Ϣ2h are the unknown constants and are given by(13)Ϣ1h=ƛ040h∗ʎhƛ400h∗ƛ040h∗−ƛ220h∗2+ƛ040h∗,and,Ϣ2h=Syh2ƛ220h∗Sxh2ʎhƛ400h∗ƛ040h∗−ƛ220h∗2+ƛ040h∗.The minimum bias and minimum MSE at the optimal values ofϢ1h and Ϣ2h are given by(14)BiasƓ3stmin=∑h=1lWh2ʎh2Syh2,ƛ040h∗ʎhƛ400h∗ƛ040h∗−ƛ220h∗2+ƛ040h∗−1,and,MSEƓ3stmin≅∑h=1lWh4ʎh3Syh4.ƛ400h∗ƛ040h∗−ƛ220h∗2ʎhƛ400h∗ƛ040h∗−ƛ220h∗2+ƛ040h∗.(v)
Singh et al. [30] proposed exponential ratio and product-type estimators which are given by(15)Ɠ4st=∑h=1lWh2ʎh2Syh2expSxh2−sxh2Sxh2+sxh2,(16)Ɠ5st=∑h=1lWh2ʎh2Syh2expSxh2−sxh2Sxh2+sxh2.The bias and MSE ofƓ4st and Ɠ5st are given by(17)BiasƓ4st=∑h=1lWh2ʎh2Syh238ƛ040h∗−12ƛ220h∗MSEƓ4st=∑h=1lWh4ʎh3Syh4ƛ400h∗+14ƛ040h∗−ƛ220h∗,and,BiasƓ5st=∑h=1lWh2ʎh2Syh212ƛ220h∗−18ƛ040h∗,(18)MSEƓ5st=∑h=1lWh4ʎh3Syh4ƛ400h∗+14ƛ040h∗+ƛ220h∗.(vi)
Shabbir and Gupta [5] proposed an estimator in stratified random sampling which is given by(19)Ɠ6st=∑h=1lWh2ʎhϢ3hsyh2+Ϣ4hSxh2−sxh2expSxh2−sxh2Sxh2+sxh2,whereϢ3h and Ϣ4h are suitably chosen constants having optimum values(20)Ϣ3hopt=ʎhƛ040h∗88−ʎhƛ040h∗ʎhƛ040h∗+ƛ004h∗ƛ040h∗−ƛ220h∗2,and,Ϣ4hopt=Syh28Sxh2−3ƛ040h∗+8ƛ220h∗2−ƛ040h∗ƛ220h∗2+4ƛ004h∗ƛ040h∗−4ƛ220h∗2ƛ040h∗+ƛ004h∗ƛ040h∗−ƛ220h∗2,BiasƓ6st≅∑h=1lWh2ʎh2Syh2+Ϣ3hSyh21+38ƛ040h∗−12ƛ220h∗+12Ϣ4hSxh2ʎhƛ040h∗.The minimum MSE ofƓ6st at the optimum values of Ϣ3hopt and Ϣ4hopt is given by(21)MSEƓ6stmin≅∑h=1lWh4ʎh3Syh464ƛ400h∗ƛ040h∗−ƛ220h∗2−ʎhƛ040h∗3−16ʎhƛ040h∗ƛ400h∗ƛ040h∗−ƛ220h∗264ʎhƛ400h∗ƛ040h∗−ƛ220h∗2+ƛ040h∗.


## 4. Proposed Estimator in Stratified Random Sampling

In survey sampling, the estimation of the finite population variance under stratified random sampling has customary very little consideration. The use of supplementary information can increase the performance of the estimator in survey sampling. When the study variable and the supplementary information are correlated with each other, then the rank of the supplementary information is also correlated with the study variable. As a result, the rank supplementary information can be observed as new supplementary information, and this information may aid in improving the efficiency of estimators. The main advantage of our proposed variance estimator under stratified random sampling is that it is more flexible and efficient than the existing estimators. Taking motivation from Shabbir and Gupta [10], we propose an improved variance estimator under stratified random sampling given by(22)Ɠpropst=∑h=1lWh2ʎhK1hsyh2+K2hSxh2−sxh2+K3hSrxh2−srxh2expSxh2−sxh2Sxh2+sxh2,(23)Ɠpropst=∑h=1lWh2ʎhK1hSyh21+ξoh−K2hSxh2ξ1h−K3hSrxh2ξ2h1−12ξ1h+38ξ1h2.After expanding (23), we have(24)Ɠpropst−∑h=1LWh2ʎh2Syh2≅∑h=1lWh2ʎh−Syh2+K1hSyh2+K1hSyh2ξoh−12K1hSyh2ξ1h−K2hSxh2ξ1h−K3hSrxh2ξ2h−12K1hSyh2ξohξ1h+38K1hSyh2ξ1h2+12K2hSxh2ξ1h2+12K3hSrxh2ξ1hξ2h.≅∑h=1lWh2ʎh−Syh2+Syh2K1h−12Syh2K1hƛ220h∗+38K1hSyh2ƛ040h∗+12Sxh2K2hƛ040h∗2+12K3hSrxh2ƛ022h∗.The bias ofƓpropst is given by(25)BiasƓpropst≅∑h=1lWh2ʎh2−Syh2+K1hSyh21+38ƛ040h∗−12ƛ220h∗+12ʎhK2hSxh2ƛ040h∗+K3hSrxh2ƛ022h∗,where K1h, K2h, and K3h are the unknown constants. The optimum values are(26)K1hopt=8−ʎhƛ040h∗ƛ400h∗81/ƛ400h∗+ʎhƔh∗+1K2hopt=Syh2ʎhƛ040h∗ƛ040h∗ƛ004h∗−ƛ220h∗2+ƛ004h∗ƛ220h∗−ƛ202h∗ƛ022h∗8−ʎhƛ040h∗+4ƛ400h∗ƛ040h∗ƛ004h−∗ƛ022h∗2−1/ƛ400h∗+ʎhƔh∗+18Sxh2ƛ400h∗ƛ040h∗ƛ004h∗−ƛ220h∗2−1/ƛ400h∗+ʎhƔh∗+1,and,K3hopt=Syh28−ʎhƛ040h∗ƛ220h∗ƛ002h∗−ƛ040h∗ƛ202h∗8Srxh2ƛ400h∗ƛ040h∗ƛ004h∗−ƛ220h∗2−1/ƛ400h∗+ʎhƔh∗+1.The minimum mean square error at the optimum values ofK1hopt, K2hopt, and K3hopt is given by(27)MSEƓpropstmin=∑h=1lWh4ʎh3Syh464Ɣh∗+1−ʎhƛ040h∗2/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1641/ƛ400h∗+ʎhƔh∗+1,where(28)Ɣh∗=2ƛ220h∗ƛ202h∗ƛ022h∗−ƛ040h∗ƛ202h∗2−ƛ004h∗ƛ220h∗2ƛ400h∗ƛ040h∗ƛ004h−∗ƛ022h∗2.


## 5. Efficiency Comparison

We compare the proposed estimator with its existing counterparts.(1)
By taking equations (5) and (27), we get(29)VarƓ0st−MSEƓpropstmin>0,∑h=1lWh4ʎh3Syh4ƛ400h∗−64Ɣh∗+1−ʎhƛ040h∗/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1641/ƛ400h∗+ʎhƔh∗+1>0.(2)
By taking (7) and (27), we get(30)MSEƓ1st−MSEƓpropstmin>0,∑h=1lWh4ʎh3Syh4ƛ400h∗+ƛ040h∗−2ƛ220h∗−64Ɣh∗+1−ʎhƛ040h∗/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1641/ƛ400h∗+ʎhƔh∗+1>0.(3)
By taking (10) and (27), we obtain(31)VarƓ2stmin−MSEƓpropstmin>0,∑h=1lWh4ʎh3Syh4ƛ400h∗1−ρh2−64Ɣh∗+1−ʎhƛ040h∗/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1641/ƛ400h∗+ʎhƔh∗+1>0.(4)
By taking (14) and (27), we obtain(32)MSEƓ3stmin−MSEƓpropstmin>0,∑h=1lWh4ʎh3Syh4ƛ400h∗ƛ040h∗−ƛ220h∗2/ʎhƛ400h∗ƛ040h∗−ƛ220h∗2+ƛ040h∗−64Ɣh∗+1−ʎhƛ040h∗/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1/641/ƛ400h∗+ʎhƔh∗+1>0.(5)
By taking (17) and (27), we obtain(33)MSEƓ4st−MSEƓpropstmin>0,∑h=1lWh4ʎh3Syh4ƛ400h∗+1/4ƛ040h∗−ƛ220h∗−64Ɣh∗+1−ʎhƛ040h∗/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1/641/ƛ400h∗+ʎhƔh∗+1>0.(6)
By taking (18) and (27), we obtain(34)MSEƓ5st−MSEƓpropstmin>0,∑h=1lWh4ʎh3Syh4ƛ400h∗+1/4ƛ040h∗−ƛ220h∗−64Ɣh∗+1−ʎhƛ040h∗/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1/641/ƛ400h∗+ʎhƔh∗+1>0.(7)
By taking (21) and (27), we obtain(35)MSEƓ6stmin−MSEƓpropstmin>0,∑h=1lWh4ʎh3Syh464ƛ400h∗ƛ040h∗−ƛ220h∗2−ʎhƛ040h∗3−16ʎhƛ040h∗ƛ400h∗ƛ040h∗−ƛ220h∗2/64ʎhƛ400h∗ƛ040h∗−ƛ220h∗2+ƛ040h∗−64Ɣh∗+1−ʎhƛ040h∗/ƛ400h∗−16ʎhƛ040h∗Ɣh∗+1/641/ƛ400h∗+ʎhƔh∗+1>0.


## 6. Numerical Study

To show the performance of our proposed estimator, we conduct a numerical study using four real data sets. We compare the performances of our proposed variance estimator with existing counterparts in terms of percentage relative efficiency. The summary statistics are given in Tables1–8. The conditional values are given in Table 9. The biases are given in Table 10. The MSEs and efficiency values are given in Tables 11 and 12. To obtain the percentage relative efficiency (PRE), we used the following expression:(36)PRE=VarƓ0stMSEƓust×100,where u=0,1,2,3,4,5,6,prop.Population 1: (Source: Murthy [31])Y: Production of a factoryX: Number of employeesRx = Rank of the X variablePopulation 2: (Source: Singh and Chaudhary [32])Y: Area under wheat in the region in 1974X: Area under wheat in the region in 1973Rx = Rank of the X variablePopulation 3: (Source: Turkey [33])Y = Apple production in 1999,X = Number of apple trees in 1999,Rx = Rank of the X variablePopulation 4: (Source: Koyuncu and Kadilar [18])Y = The number of teachersX = The number of classes in both primary and secondary schools in Turkey in 2007 for 923 districts in six regionsRx = Rank of the X variableTable 1
Summary statistics using population 1.
hNhnhʎhWhY¯hX¯hR¯xhSyh2119100.0480.23752975.2664.8910583977.5232160.0320.44653.28140.1216456563.331470.0800.43756553.28402.087.5195208.841580.0600.1875782.66764.278437923.5Table 2
Summary statistics using population 1.
ƛ400h∗ƛ040h∗ƛ004h∗ƛ220h∗ƛ202h∗ƛ022h∗2.280.450.690.460.780.530.562.090.750.740.580.980.620.620.660.800.780.631.220.900.671.020.700.69Table 3
Summary statistics using population 2.
hNhnhʎhWhY¯hX¯hR¯xhSyh21930.2220.2647253253.4531978.2521030.230.2941213.5226.85.537629.3931540.180.4411157.9170.286893.067Table 4
Summary statistics using population 2.
ƛ400h∗ƛ040h∗ƛ004h∗ƛ220h∗ƛ202h∗ƛ022h∗1.92861.070.571.380.770.640.5110.420.590.420.410.401.201.420.671.280.650.76Table 5
Summary statistics using population 3.
hNhnhʎhWhY¯hX¯hR¯xhSyh2110690.101670.124121536.7724375.553.45412817462106170.049380.124122212.5927421.753.33133437791394380.015670.110079384.3072409.947.448944574334171670.009070.200235588.0174364.685.88820445636520470.137950.23887966.9526441.7102.305710999617320.494210.20257404.399843.886.8089440.3Table 6
Summary statistics using population 3.
ʎ400h∗ʎ040h∗ʎ004h∗ʎ220h∗ʎ202h∗ʎ022h∗76.850525.94040.765831.98621.73591.58083693.270934.23250.766555.85541.87551.58285424.263326.11690.762919.58221.63951.56909899.789895.59530.780297.94521.86031.82750353.512828.48380.783519.98861.60951.60371228.772428.91540.781221.94651.52071.579165Table 7
Summary statistics using population 4.
hNhnhʎhWhY¯hX¯hR¯xhSyh21127310.024380.13759703.74020804.5964.00781163.92117210.039070.12676413.0009211.7959.00415924.83103290.024770.11159573.17414309.3052.001068054.04170380.020430.18418424.6649478.8585.48657047.85205220.040570.22210267.0295569.94102.99162936.96201390.200660.21177393.84012997.59100.99506549Table 8
Summary statistics using population 4.
ʎ400h∗ʎ040h∗ʎ004h∗ʎ220h∗ʎ202h∗ʎ022h∗3.947836.2515890.7716183.7204880.95879471.18810217.3318119.356220.76918918.352091.5036091.51701515.8713616.30730.76499616.090881.5156921.52505113.6037511.679990.77886211.656051.446321.43127622.3190823.148650.78222722.300211.512241.52691721.4988224.260140.78176621.793861.5723951.601076Table 9
Conditional values of different estimators using real data sets.
Conditional valuesPopulation-IPopulation-IIPopulation-IIIPopulation-IVCondition I308077.037170053.88236185082878990268317.021Condition II288895.33724619.252364076397395927349.521Condition III144739.5372754.814363144452812722541.981Condition IV127762.3372238.92116349115287999365.981Condition V293935.63755621.4257883019573416102341.721Condition VI649319.337351276.6823688294916731146686.721Condition VII125488.04451040.4690613925506134265.953Table 10
Biases values of different estimators using real data sets.
EstimatorsPopulation-IPopulation-IIPopulation-IIIPopulation-IVƓ0st————Ɠ1st83.9204−30.89047−137147.834.49139Ɠ2st————Ɠ3st254.9−1354.09619.748037.871472Ɠ4st11.00481−54.05082−152436.6−44.53858Ɠ5st181.1137317.4172917899.3595.1742Ɠ6st182588.8169013.421.8469903e − 11264051.068Ɠpropst308077.037170053.89163411187990268317.021Table 11
Mean square error (MSE) of estimators using populations I–IV.
EstimatorsPopulation-IPopulation-IIPopulation-IIIPopulation-IVƓ0st425574188538187490767203272821.8Ɠ1st406392.326463.374317186217231854.3Ɠ2st262236.521238.93203385241634027046.76Ɠ3st245259.320723.03883515231701213870.76Ɠ4st305432.674105.5434285427461629106846.5Ɠ5st766816.3369760.890702804944151191.5Ɠ6st242985.2007519524.58670128004388268359.251Ɠpropst117496.96318484.1176424078882134504.779Table 12
Percentage relative efficiency (PRE) of estimators using populations I–IV.
EstimatorsPopulation-IPopulation-IIPopulation-IIIPopulation-IVƓ0st100100100100Ɠ1st104.724712.430434.2893856.4677Ɠ2st162.280887.700553.84751008.704Ɠ3st173.523909.799219.47361966.885Ɠ4st139.333254.412206.7089255.34Ɠ5st55.49881208.2466695.05180.4478Ɠ6st175.144965.6445293.6443263.711Ɠpropst362.2001020.007786.5236056.276


## 7. Simulation Study

A simulation study is performed to determine the efficiency of the estimators for variance stratified random sampling. Three populations are generated from normal distribution by using theR language program. The first population is generated for equal strata and the second one is generated for unequal strata, and the third one is generated for equal strata of a small sample size. The population details are given as follows.


### 7.1. Population I

N1=1000,N2=1000,N3=1000,N4=1000,N5=1000,N6=1000,n1=200,n2=200,n3=200,n4=200,n5=200,n6=200,X1=rnorm\1000,5,10,U1=X1+rnorm\1000,0,1,Y1=U1+rnorm\1000,1,3,X2=rnorm\1000,4,8,U2=X2+rnorm\1000,0,1,Y2=U2+rnorm\1000,1,3,X3=rnorm\1000,4,9,U3=X3+rnorm\1000,0,1,Y3=U3+rnorm\1000,1,3,X4=rnorm\1000,3,7,U4=X4+rnorm\1000,0,1,Y5=U5+rnorm\1000,1,3,X5=rnorm\1000,3,8,U5=X5+rnorm\1000,0,1,Y5=U5+rnorm\1000,1,3,X6=rnorm\1000,2,5,U6=X6+rnorm\1000,0,1,Y6=U6+rnorm\1000,1,3.


### 7.2. Population II

N1=1200,N2=800,N3=1300,N4=900,N5=1100,N6=700,n1=240,n2=160,n3=260,n4=180,n5=220,n6=140,X1=rnorm\1200,5,10,U1=X1+rnorm\1200,0,1,Y1=U1+rnorm\1200,1,3,X2=rnorm\800,4,8,U2=X2+rnorm\800,0,1,Y2=U2+rnorm\800,1,3,X3=rnorm\1300,4,9,U3=X3+rnorm\1300,0,1,Y3=U3+rnorm\1300,1,3,X4=rnorm\900,3,7,U4=X4+rnorm\900,0,1,Y5=U5+rnorm\900,1,3,X5=rnorm\1100,3,8,U5=X5+rnorm\1100,0,1,Y5=U5+rnorm\1100,1,3,X6=rnorm\700,2,5,U6=X6+rnorm\700,0,1,Y6=U6+rnorm\700,1,3.


### 7.3. Population III

N1=300,N2=300,N3=300,N4=300,N5=300,N6=300,n1=60,n2=60,n3=60,n4=60,n5=60,n6=60,X1=rnorm\300,5,10,U1=X1+rnorm\300,0,1,Y1=U1+rnorm\300,1,3,X2=rnorm\300,4,8,U2=X2+rnorm\300,0,1,Y2=U2+rnorm\300,1,3,X3=rnorm\300,4,9,U3=X3+rnorm\300,0,1,Y3=U3+rnorm\300,1,3,X4=rnorm\300,3,7,U4=X4+rnorm\300,0,1,Y5=U5+rnorm\300,1,3,X5=rnorm\300,3,8,U5=X5+rnorm\300,0,1,Y5=U5+rnorm\300,1,3,X6=rnorm\300,2,5,U6=X6+rnorm\300,0,1,Y6=U6+rnorm\300,1,3.The percentage relative efficiency (PRE) is calculated as follows:(37)PRE=VarƓ0stMSEƓjst∗100,where j=0,1,2,3,4,5,6, prop.The mean square error and percentage relative efficiency values are given in Tables13 and 14.Table 13
Mean square error (MSE) of estimators based on the simulation study.
EstimatorsPopulation-IPopulation-IIPopulation-IIIƓ0st5.167166e − 067.274847e − 060.0001922386Ɠ1st3.510773e − 064.860487e − 060.0001276158Ɠ2st2.723914e − 063.727381e − 060.0001027585Ɠ3st1.750417e − 062.244754e − 066.313865e − 05Ɠ4st3.667552e − 064.874891e − 060.000141089Ɠ5st9.072873e − 061.1539e − 050.0003547078Ɠ6st3.005942e − 073.781101e − 071.031059e − 05Ɠpropst7.132623e − 087.55517e − 082.195092e − 06Table 14
Percentage relative efficiency (PRE) of estimators based on the simulation study.
EstimatorsPopulation-IPopulation-IIPopulation-IIIƓ0st100100100Ɠ1st147.1803149.6732150.6385Ɠ2st189.6963195.1732187.0781Ɠ3st295.1963324.0821304.4705Ɠ4st140.8887149.231136.2534Ɠ5st56.9518163.0457254.19633Ɠ6st1718.9841924.0021864.477Ɠpropst7244.4119628.9648757.656


## 7.1. Population I

N1=1000,N2=1000,N3=1000,N4=1000,N5=1000,N6=1000,n1=200,n2=200,n3=200,n4=200,n5=200,n6=200,X1=rnorm\1000,5,10,U1=X1+rnorm\1000,0,1,Y1=U1+rnorm\1000,1,3,X2=rnorm\1000,4,8,U2=X2+rnorm\1000,0,1,Y2=U2+rnorm\1000,1,3,X3=rnorm\1000,4,9,U3=X3+rnorm\1000,0,1,Y3=U3+rnorm\1000,1,3,X4=rnorm\1000,3,7,U4=X4+rnorm\1000,0,1,Y5=U5+rnorm\1000,1,3,X5=rnorm\1000,3,8,U5=X5+rnorm\1000,0,1,Y5=U5+rnorm\1000,1,3,X6=rnorm\1000,2,5,U6=X6+rnorm\1000,0,1,Y6=U6+rnorm\1000,1,3.


## 7.2. Population II

N1=1200,N2=800,N3=1300,N4=900,N5=1100,N6=700,n1=240,n2=160,n3=260,n4=180,n5=220,n6=140,X1=rnorm\1200,5,10,U1=X1+rnorm\1200,0,1,Y1=U1+rnorm\1200,1,3,X2=rnorm\800,4,8,U2=X2+rnorm\800,0,1,Y2=U2+rnorm\800,1,3,X3=rnorm\1300,4,9,U3=X3+rnorm\1300,0,1,Y3=U3+rnorm\1300,1,3,X4=rnorm\900,3,7,U4=X4+rnorm\900,0,1,Y5=U5+rnorm\900,1,3,X5=rnorm\1100,3,8,U5=X5+rnorm\1100,0,1,Y5=U5+rnorm\1100,1,3,X6=rnorm\700,2,5,U6=X6+rnorm\700,0,1,Y6=U6+rnorm\700,1,3.


## 7.3. Population III

N1=300,N2=300,N3=300,N4=300,N5=300,N6=300,n1=60,n2=60,n3=60,n4=60,n5=60,n6=60,X1=rnorm\300,5,10,U1=X1+rnorm\300,0,1,Y1=U1+rnorm\300,1,3,X2=rnorm\300,4,8,U2=X2+rnorm\300,0,1,Y2=U2+rnorm\300,1,3,X3=rnorm\300,4,9,U3=X3+rnorm\300,0,1,Y3=U3+rnorm\300,1,3,X4=rnorm\300,3,7,U4=X4+rnorm\300,0,1,Y5=U5+rnorm\300,1,3,X5=rnorm\300,3,8,U5=X5+rnorm\300,0,1,Y5=U5+rnorm\300,1,3,X6=rnorm\300,2,5,U6=X6+rnorm\300,0,1,Y6=U6+rnorm\300,1,3.The percentage relative efficiency (PRE) is calculated as follows:(37)PRE=VarƓ0stMSEƓjst∗100,where j=0,1,2,3,4,5,6, prop.The mean square error and percentage relative efficiency values are given in Tables13 and 14.Table 13
Mean square error (MSE) of estimators based on the simulation study.
EstimatorsPopulation-IPopulation-IIPopulation-IIIƓ0st5.167166e − 067.274847e − 060.0001922386Ɠ1st3.510773e − 064.860487e − 060.0001276158Ɠ2st2.723914e − 063.727381e − 060.0001027585Ɠ3st1.750417e − 062.244754e − 066.313865e − 05Ɠ4st3.667552e − 064.874891e − 060.000141089Ɠ5st9.072873e − 061.1539e − 050.0003547078Ɠ6st3.005942e − 073.781101e − 071.031059e − 05Ɠpropst7.132623e − 087.55517e − 082.195092e − 06Table 14
Percentage relative efficiency (PRE) of estimators based on the simulation study.
EstimatorsPopulation-IPopulation-IIPopulation-IIIƓ0st100100100Ɠ1st147.1803149.6732150.6385Ɠ2st189.6963195.1732187.0781Ɠ3st295.1963324.0821304.4705Ɠ4st140.8887149.231136.2534Ɠ5st56.9518163.0457254.19633Ɠ6st1718.9841924.0021864.477Ɠpropst7244.4119628.9648757.656


## 8. Discussion

We used four real data sets to obtain the percentage relative efficiency of all estimators under variance stratified random sampling. The data descriptions of these populations are presented in Tables1–8. All the conditional values of different estimators using four real data sets are given in Table 9. Bias values of the proposed and existing estimators using real data sets are presented in Table 10. The mean square error and percentage relative efficiency of estimators based on real data sets are given in Tables 11 and 12. Similarly, MSE and PRE of the simulation study are given in Tables 13 and 14. From the numerical results, it is observed that the proposed estimator is appreciable in terms of the minimum mean square error and higher percentage relative efficiency (PRE) as compared to existing estimators. The larger gain in efficiency is observed by using the proposed estimator over some existing estimators under stratified random sampling. The results integrated with this study are very sound and quite illuminating. Thus, it is recommended that the proposed estimator is useful in practice.


## 9. Conclusion

Several estimators for estimating the finite population variances are constructed on the basis of the auxiliary variable. We proposed an improved estimator using the dual supplementary information for a variance estimator in stratified sampling. The proposed estimator is the particular class of estimators given by Shabbir and Gupta [10]. The bias and MSE of the existing and proposed estimator are derived up to the first order of approximation. The performance of the proposed estimators is the best as compared to existing counterparts in terms of efficiency. Four real data sets are used to assess the efficiency of the proposed estimator over its existing counterparts. Moreover, a simulation study is also carried out to check the robustness and generalizability of the proposed variance estimator. Based on the numerical findings, the proposed estimator may be preferable for use in practical situations. The possible extension of this current work is to develop an improved class of estimators under nonresponse, distribution function, and measurement error for estimating the finite population variance under stratified random sampling.



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*Source: 3813952-2022-11-17.xml*