This article deals with estimation of finite population mean using the auxiliary proportion under simple and two phase sampling scheme utilizing two auxiliary variables. Mathematical expressions for the mean squared errors of the proposed estimators are derived under first order of approximation. We compare the proposed class of estimators "theoretically and numerically" with the usual mean estimator of Naik and Gupta [1]. The theoretical as well as numerical findings support the superiority of our proposed class of estimator as compared to estimators available in literature.
Citation: Xuechen Liu, Muhammad Arslan. A general class of estimators on estimating population mean using the auxiliary proportions under simple and two phase sampling[J]. AIMS Mathematics, 2021, 6(12): 13592-13607. doi: 10.3934/math.2021790
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Abstract
This article deals with estimation of finite population mean using the auxiliary proportion under simple and two phase sampling scheme utilizing two auxiliary variables. Mathematical expressions for the mean squared errors of the proposed estimators are derived under first order of approximation. We compare the proposed class of estimators "theoretically and numerically" with the usual mean estimator of Naik and Gupta [1]. The theoretical as well as numerical findings support the superiority of our proposed class of estimator as compared to estimators available in literature.
1.
Introduction
1.1. Introduction and notation in simple random sampling
It is a well-known fact, that at large scale survey sampling, the use of several auxiliary variables improve the precision of the estimators. In survey sampling, researchers have already attempted to obtain the estimates for population parameter such as mean, median etc, that posses maximum statistical properties. For that purpose a representative part of population is needed, when population of interest is homogeneous then one can use simple random sampling (SRS) for selecting units. In some situations, information available in the form of attributes, which is positively correlated with study variables. Several authors including Naik and Gupta [1], Jhajj [2], Abd-Elfattah [3], Koyuncu [4], Solanki [5], Sharma [6] and Malik [7] proposed a set of estimators, taking the advantages of bi-serial correlation between auxiliary and study variables, utilizing information on single auxiliary attribute. Verma [8], Malik [7], Solanki et al., [9] and Sharma [10] suggested some estimators utilizing information on two auxiliary attributes in SRS, Mahdizadeh and Zamanzade [11] developed a kernel-based estimation of P(X>Y) in ranked-set sampling, SinghPal and Solanki [12] developed a new class of estimators of finite population mean survey sampling and Mahdizadeh and Zamanzade [13] suggest a smooth estimation of a reliability function in ranked set sampling, further more Hussain et al., [14] and Al-Marzouki et al., [15] also work in this side.
In this article, we consider the problem of estimating the finite population mean using the auxiliary proportion under simple and two phase sampling scheme. The mathematical expression of the bias and mean squared error of the proposed estimator are derived under first order of approximation. The performance of proposed class of estimator is compared with that of the existing estimators both theoretically and numerically. In terms of percentage relative efficiency (PRE), it is found that proposed class of estimator outperforms the existing ones.
Let U={u1,u2,...,uN} represent a finite population of size N distinct units, assumed that a sample of size n units is drawn from this population U using simple random sampling without replacement. Let yi and ϕij (i = 1, 2) denotes the observations on variable y and ϕi (i = 1, 2) for the jth unit (j = 1, 2, ..., N).
ϕij=1, if ith unit posses atrributes
ϕij=0, otherwise
Pj=∑Niϕij=Aj/N,(j=1,2) and pj=∑Niϕij=aj/n,(j=1,2) are the population and sample proportions of auxiliary variable respectively. Let ˉY=∑Ni=1yiN, ˉy=∑ni=1yin be the population and sample mean of the study variable y. S2ϕjy=∑Ni=1(ϕij−Pj)(yi−¯Y)N−1,(j=1,2) are the variations between the study and the auxiliary attributes. S2ϕ1ϕ2=∑Ni=1(ϕi1−P1)(ϕi2−P2)N−1 are the variations between the auxiliary attributes. ρyϕj=SϕjySySϕ represents the point bi-serial correlation between the study variable y and the two auxiliary attributes p1 and p2 respectively. ρϕ1ϕ2=Sϕ1ϕ2Sϕ1Sϕ2 represents the point bi-serial correlation between the two auxiliary attributes p1 and p2 respectively.
Let us define, e0=ˉy−ˉYˉY, e1=p1−P1P1, e2=p2−P2P2,
Where Cy=SyˉY, Cϕj=SϕjPj,(j=1,2), is the co-efficient of variation of the study and auxiliary attribute. S2y=∑Ni=1(yi−ˉY)2N−1, S2ϕj=∑Ni=1(ϕij−Pj)2N−1,(j=1,2), is the variance of study and auxiliary attribute. f=(1n−1N) is the correction factor.
The rest of the paper is organized as follows. In Sections 1.1 and 1.2, introduction and notations are given for simple random sampling and two phase sampling. In Sections 2.1 and 2.3, we discussed some existing estimators of the finite population mean for both sampling designs. The proposed estimators are given in Sections 2.2 and 2.4. In Sections 3.1 and 3.2, theoretical comparisons are conducted. While in Sections 4.1 and 4.2 we focus on empirical studies. Finally, application and conclusions are drawn in Sections 5 and 6.
1.2. Introduction and notation in two phase sampling
The precision of estimate can be increased by using two methodologies. Firstly the precision may be increased by using using adequate sampling design for the estimated variable. Secondly the precision may be increased by using an appropriate estimation procedure, i.e. some auxiliary information which is closely associated with the variable under study. In application there exist a situation when complete auxiliary information or attribute is not available or information on that attribute is expensive. In that case, a method of two phase sampling or double sampling is used to obtain the estimates of unknown population parameters. In two phase sampling, a large preliminary sample (n′) is selected by SRSWOR to obtain the estimate of unknown parameter of the auxiliary variable at first phase and the information on the auxiliary variable is collected, which is use to estimate the unknown auxiliary variable. Then a sub sample (n<n′) is selected at second phase and both the study and auxiliary variables are collected. Here we assume that Population proportion (P1) is unknown and introduce an improved estimator to estimate the population mean. Kiregyera [16], Mohanty [17], Malik [7] and Haq [18] used two auxiliary variables in two phase sampling for the better estimation of mean.
An example in this context is while estimating the yield of a crop, it is likely that the area under the crop may be unknown but the area of each farm may be known. Then y, P1 and P2 respectively are the yield area under the crop and area under cultivation.
Consider a finite population U=(u1,u2...uN) of size N and let yi, ϕi1 and ϕi2 is the information on the study variable and two auxiliary attributes associated with each unit ui(i=1,2,...,N) of the population such that:
ϕij=1, if the ith unit in the population possesses auxiliary attribute ϕj, ϕij=0 otherwise.
We assume that the population mean of the first auxiliary proportion P1 is unknown but the same information is known for the second proportion. Let pj′=∑n′iϕijn=aj/n′ for j=1,2 be the estimate of Pj obtained from the first phase sample of size n′, drawn by using SRSWOR from the population of N units. Let ˉy=∑niyi and p1=∑niϕi1n=a1/n be the estimates of ˉY and P1 respectively, obtained from a second sample of size n, drawn from the first phase n′ using SRSWOR.
To obtain the bias and MSE for estimators in two phase sampling we define the error terms as follows:
s2ϕj=∑ni=1(ϕij−pj)2n−1, represents the sample variance of size n,
s2ϕj′=∑n′i=1(ϕij−pj)2n′−1, represents the sample variance of size n′
ρyϕj=SϕjySySϕ represent point bi-serial correlation between the study variable (y) and the two auxiliary attributes (P1) and (p2).
ρϕ1ϕ2=Sϕ1ϕ2Sϕ1Sϕ2 represent point bi-serial correlation between the two auxiliary attributes (P1) and (P2) respectively.
2.
Existing and proposed estimators
2.1. Existing estimators in simple random sampling
In order to have an estimate of the study variable, using information of population proportion P, Naik [1] proposed the following estimators respectively.
where b1=syϕ1s2ϕ1 and b2=syϕ2s2ϕ2 are the sample regression coefficients. γ1 and γ2 are two unknown constants. The optimum values of these constants are given as:
where k1 and k2 are suitable constants whose values are to be determined such that MSE of tRPR is minimum; η and λ are either real numbers or functions of known parameters of the auxiliary attribute ϕ2 such as coefficient of variation (Cϕ2), coefficient of kurtosis (βϕ2) and α is the scalar (0≤α≤1) for designing different estimators. Let ˉY and (P1,P2) be the population means of the study variable and auxiliary proportions respectively. ˉy and (p1,p2) be the sample means of the study variable and auxiliary proportions respectively.
Putting α=1 and α=0 in (2.13), we get the following estimators.
For α=1, the suggested class of estimators reduces to:
The optimum values of k1 and k2 are obtained by minimizing Eq (2.18) and is given by
k1=BE−2CD2(AE−C2),
and
k2=ˉY(2AD−BC)2(AE−C2),
Substituting the optimum values of k1 and k2 in Eq (2.18) we get the minimum MSE of tRPR as:
MSE(t(RPR)min)=ˉY2(1−4AD2+B2E−4BCD)4(AE−C2).
(2.19)
The minimum MSE of the proposed estimator tRPR at Eq (2.19) depends upon many parametric constants, we use these constant for readers to easily understand and for notation convenient.
2.3. Existing estimators in two phase sampling
The usual mean per unit estimator in two phase sampling is:
t′U=ˉy.
(2.20)
The MSE of t′U is given by
MSE(t′U)=ˉY2V200.
(2.21)
The Naik [1] estimators in two phase sampling are :
tA′=ˉy(p′1p1),
(2.22)
tB′=ˉy(P2p′2),
(2.23)
tC′=ˉyexp(p1′−p1p1′+p1),
(2.24)
tD′=ˉyexp(P2−p2′P2+p2′).
(2.25)
The MSE expressions of estimators tA′, tB′, tC′ and tD′ are respectively given as:
MSE(tA′)≅ˉY2(V200+V020−V′020+2V′110−2V′110),
(2.26)
MSE(tB′)≅ˉY2(V200+V′002+2V′101),
(2.27)
MSE(tC′)≅ˉY2(V200+V′110−V110−14V′020+14V020),
(2.28)
MSE(tD′)≅ˉY2(V200+14V′002+V′101).
(2.29)
Malik [7] used exponential type estimator with regression coefficients in two phase sampling which is given by:
where b1=syϕ1s2ϕ1 and b2=syϕ2s2ϕ2 are the sample regression coefficients. δ1 and δ2 are two unknown constants. The optimum values of these constants are given as:
δ1(opt)=−2P1β1ˉY+2Cyρyϕ1Cϕ1,
δ2(opt)=−2P2β2ˉY+2Cyρyϕ2Cϕ2,
where, β1=Syϕ1S2ϕ1 and β2=Syϕ2S2ϕ2 are the regression coefficients.
The minimum mean square error for the optimum values of δ1 and δ2 are given as:
MSE(tMSmin)≅fˉY2C2y{−f(−1+ρ2yϕ1)+λ(ρ2yϕ1−ρ2yϕ2)}.
(2.31)
2.4. A generalized proposed class of estimators in two phase sampling
We suggest a generalized exponential estimator when P1 is unknown and P2 is known:
where k′1 and k′2 are suitable constants whose value are to be determined such that MSE of tRPR is minimum. η′ and λ′ are either real numbers or functions of known parameters of the auxiliary attribute ϕ2 such as coefficient of variation, coefficient of kurtosis (βϕ2) and α′ is a scalar (0≤α≤) for designing different estimators.
Putting α′=1 and α′=0 in above suggested class of estimators, we get the following estimators.
For α′=1, the suggested class of estimators reduces to:
We use the following expression to obtain the Percentage Relative Efficiency PRE:
PRE=MSE(t0)MSE(timin)∗100,
(4.1)
where i = U, A, B, C, D, MS, RPR1, RPR2, RPR3, RPR4, RPR5, RPR6, RPR7 and RPR8.
In Table 2, it is clearly shown that our suggested class of estimator tRPRi perform better than all the existing estimators tA, tB, tC, tD and tMS. A significant increase is observed in the percentage relative efficiency of estimators of tRPR6, tRPR7 and tRPR8.
Table 2.
Set of estimators generated from estimator t′RPR(α′=1).
We use the following expression to obtain the Percentage Relative Efficiency(PRE):
PRE=MSE(t0)MSE(t′imin)∗100,
(4.2)
where i = U′, A′,B′,C′,D′,MS′,RPR1′,RPR2′,RPR3′,RPR4′,RPR5′,RPR6′,RPR7′andRPR8′.
The results for data set 1–4 are given in Table 4.
In Table 4, it is clearly shown that our suggested class of estimator t′RPR perform better than all the existing estimators of t′A, t′B, t′C and t′D and t′MS. A significant increase is observed in the percentage relative efficiency of estimators of t′RPR6, t′RPR7 and t′RPR8.
There are many situations where we only interest in knowing everything about the study variable, which is too difficult. For this we can use two auxiliary variables in the form of proportion to find out the study variable. This manuscript provides us the basic tools to the problems related to proportion estimation and two-phase sampling. Here we can see that in abstract of the manuscript we just talk about the minimum MSE of proposed and existing estimators, reason behind is that we can easily compare the minimum MSE with other properties of good estimators like MLE ect., we can also see that the comparison is made in the form of percentage relative efficiency.
5.
Application
Statisticians are constantly trying to develop efficient estimators and estimation methodologies to increase the efficiency of estimates. The progress is going on for estimators of population mean. In the present paper our task is to develop a new estimator for estimating the finite population mean under two different sampling schemes, which are simple random sampling and two-phase sampling. The new estimators will be proposed under the following situations:
1). The initial sample is collected through simple random sampling.
2). And then by two-phase sampling using simple random sampling.
In this article, we consider the problem of estimating the finite population mean using the auxiliary proportion under simple random sampling and two-phase sampling scheme. In general, during surveys, it is observed that information in most cases is not obtained on the first attempt even after some call-backs, in such types of issue we use simple random sampling. And when the required results are not obtained, we use two-phase sampling. These approaches are used to obtain the information as much as possible. In sample surveys, it is well known that while estimating the population parameters, i.e., Finite population (mean, median, quartiles, coefficient of variation and distribution function) the information of the auxiliary variable (Proportion) is usually used to improve the efficiency of the estimators. The main aim of studies is to find out more efficient estimators than classical and recent proposed estimators using the auxiliary information (in the form of proportion) for estimating finite population mean under simple random sampling and two-phase sampling scheme.
There are situations where our work is deemed necessary and can be used in daily life.
1). For a nutritionist, it is interesting to know the proportion of population that consumes 25% or more of the calorie intake from saturated fat.
2). Similarly, a soil scientist may be interested in estimating the distribution of clay percent in the soil.
3). In addition, policy-makers may be interested in knowing the proportion of people living in a developing country below the poverty line.
6.
Conclusions
In this paper, we have proposed a generalized class of exponential ratio type estimators for estimating population mean using the auxiliary information in the form of proportions under simple and two phase sampling. We used SRS to estimate the population mean using the proportions of available auxiliary information, and when the auxiliary information is unknown, we used two phase sampling for estimation resolution. From the numerical results available in Tables 3 and 4 we can see that two phase sampling gave more efficient results than simple random sampling. Thus the use of auxiliary information in estimation processes increases the efficiency of the estimator, that's we have used two auxiliary variables as attributes. In the numerical study we showed that the proposed estimator is more efficient that tU, tA, tB, tC, tD, tMS and any other suggested family of estimators both in simple and two phase sampling schemes.
Table 3.
Percentage relative efficiency (PRE) with respect to usual mean estimator tU.
Some possible extensions of the current work are as follows:
Develop improved finite population mean estimators,
1). using supplementary information more than one auxiliary variable.
2). under stratified two-phase sampling.
3). in the presence of measurement errors.
4). under non-response with two-phase sampling.
Acknowledgments
The authors are thankful to the learned referee for his useful comments and suggestions.
Conflict of interest
The authors declare no conflict of interest.
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Xuechen Liu, Muhammad Arslan. A general class of estimators on estimating population mean using the auxiliary proportions under simple and two phase sampling[J]. AIMS Mathematics, 2021, 6(12): 13592-13607. doi: 10.3934/math.2021790
Xuechen Liu, Muhammad Arslan. A general class of estimators on estimating population mean using the auxiliary proportions under simple and two phase sampling[J]. AIMS Mathematics, 2021, 6(12): 13592-13607. doi: 10.3934/math.2021790