# Improved Estimators of Population Mean under Nonresponse in Successive Sampling

Authors: Prayas Sharma; Surya K. Pal; Housila P. Singh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

It is well accepted that suitable additional information improves the efficiency of an estimator but at the same time, it faced the situation of nonresponse. The two-occasion successive sampling is useful to handle the absence of a full response from respondents. In this present study, we have developed exponential estimators for population mean using a subsampling nonrespondent procedure. To show the efficacy of the recommended estimators, several properties are derived, and respective optimum replacement strategies are inferred. To evaluate the performance of the recommended estimators empirically, a computational study is carried out as well. It was found that the recommended estimators outperform the existing ones under the nonresponse in successive sampling.


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## Body

## 1. Introduction

Incomplete information is a well-known problem in sample surveys, especially in socio-economic surveys of households, in which individual data are collected. The reasons for missing information may be migration, refusal to respond, not being available at the time of surveys performed, etc.Jessen [1] initially encountered the problem and suggested the method of estimation under the successive sampling with partial replacement of units utilizing the complete information at the previous occasion. Furthermore, Patterson [2]; Rao and Graham [3]; Feng and Zou [4] studied the properties of different estimators under successive sampling. Biradar and Singh [5] and Singh and Vishwakarma [6]; Singh and Pal [7]; Sanahulla et al. [8]; Javaid et al. [9] and Pal et al. [10] used additional information for estimation under successive sampling.The concept used in this paper is nonresponse. First time during the data collection, Hansen and Hurwitz [11] realize the problem of nonresponse at the estimation stage. He took this problem forward and realizes that while taking the subsamples from the nonrespondents group effect of nonresponse can be reduced. Furthermore, he developed the estimators utilizing the information from response and nonresponse groups together which was well accepted in the sampling theory.Later on, several researchers including Chaudhary et al. [12]; Singh and Priyanka [13], and Pal and Singh [14] combined the concept of successive sampling and nonresponse and used it for estimation of population mean on current occasion on two-occasion successive sampling. Under different practical situations using the auxiliary variable, the aforesaid authors suggested some estimators when nonresponse was observed on the current occasion in two-occasion successive sampling.The remainder of this paper is organized as follows: in section2, terminology and notations are presented further the resent estimators and proposed estimators along with their properties are in sections 3 and 4, respectively. Section 5 provides the computational study, and concluding remarks are offered in section 6.


## 2. Terminology and Notations

LetΩ=Ω1,Ω2,…,ΩN be a finite population of N units, which has been sampled over two occasions. The character under study is denoted by xy on the first (second) occasion. It is assumed that information on an ancillary variable z (with unknown population mean), which is positively correlated with the study variable, is readily available and almost stable over both the occasions. A simple random sample (without replacement) sn of n units is drawn on the first occasion. A random subsample sm of m=nλ units is retained (matched) for its use on the second occasion. We assume that there is nonresponse at the current occasion, so that the population can be divided into two classes, those who will respond at the first attempt and those who will not respond. Let the sizes of these two classes be N1 and N2, respectively. At the current (second) occasion, a simple random sample (without replacement) sυ of υ=n−m=nμ units is drawn afresh from the entire population so that the sample size on the current (second) occasion is also n. λ and μ, λ+μ=1 are the fractions of matched and fresh samples, respectively, at the current (second) occasion. We assume that in the unmatched portion of the sample on the current (second) occasion, υ1 units respond and υ2 units do not respond. Let sυ1∩sυ and sυ2∩sυ. Let υ2h denote the size of the subsample sυ2h (of sυ2 drawn from the nonresponding units in the unmatched (fresh) portion sυ2 of the sample sυ on the current (second) occasion (i.e., from the υ2 nonrespondents, on SRSWOR of υ2h units is selected with the inverse sampling rates f2, where υ2h=υ2/f2, f2>1.The notations used in the research paper are shown in the Table1.


## 3. Recent Developments of Estimators

For estimating the current population meanY¯, Singh et al. [15, 16] have given the estimators of set sυ as(1)T1υ=y¯υ∗expZ¯−z¯υ∗Z¯+z¯υ,T2υ=y¯υ∗expZ¯−z¯υZ¯+z¯υ∗.Singh et al. [15] also gave the estimators of set sm for estimating the current population mean Y¯ as(2)Tm=y¯mZ¯z¯mexpx¯n−x¯mx¯n+x¯m.Singh et al. [15] suggested the following estimators of population mean Y¯ at the current (second) occasion by combining the estimators of sets sυ and sm as(3)Ti=ϕiTiu+1−ϕiTim;i=1,2,where ϕi0≤ϕi≤1; i=1,2 are unknown constants (scalars) to be determined under certain criteria. For detailed properties of the estimators Tii=1,2, readers are referred to Singh et al. [15].


## 4. New Development of Estimators

### 4.1. Estimators Based on the Unmatched Portion of the Amplesu

Looking the formulation of the estimatorsT1υ given by (1) due to Singh et al. [15], it is clear that the estimator T1υ is defined in the situation, in which information on the auxiliary variable z is obtained for all the sample units υ (drawn afresh from the entire population at the second occasion), and the population mean Z¯ of the auxiliary variable z is known, but some sample units fail to supply information on the study variable y. We note that, when suggesting the estimators for the population mean Y¯ on the current (second) occasion, Singh et al. [15] used only information on the sample mean z¯υ of the auxiliary variable z. However, one can also obtain the unbiased estimator z¯υ∗ of the population mean Z¯ (without any extra effort) while in the process of obtaining the unbiased estimator y¯υ∗ of the population mean Y¯. Thus, in this situation, where information on the auxiliary variable z is obtained for all the sample units υ, we have two unbiased estimators z¯υ∗ and z¯υ, of the population mean Z¯ of the auxiliary variable z (see, Singh and Kumar [17]). With this background, we suggest the following estimators (using z¯υ∗ and z¯υ together) for the population mean Y¯ on the current (second) occasion based on unmatched portion of the sample Sυ of the size υ.(4)P1υ=y¯υ∗expZ¯−z¯υZ¯+z¯υexpZ¯−z¯υ∗Z¯+z¯υ∗,P2υ=y¯υ∗exp2β^yzR^zZ¯−z¯υZ¯+z¯υexp4β^yz2R^zZ¯z¯υ−z¯υ∗Z¯+z¯υZ¯+z¯υ∗,where β^yz∗=syz∗/sz∗2,β^yz2∗=syz2∗/sz2∗2 are, respectively, the estimates of βyz2∗,βyz2∗,(5)syz∗=1υ−1∑sυ1yjzj+f∑sυ2hyjzj−υy¯υ∗z¯υ∗,sz∗2=1υ−1∑sυ1zj2+f∑sυ2hzj2−υz¯υ∗2,syz2∗=1υ2h−1∑sυ2hyj−y¯υ2hzj−z¯υ2h,sz2∗2=1υ2h−1∑sυ2hzj−z¯υ2h2,R^z=y¯υ∗Z¯,y¯υ2h=1υ2h∑sυ2hyj,z¯υ2h=1υ2h∑sυ2hzj.For the simplicity in calculation, Reddy (1974, [18]) and Ruiz Espejo [19] initially assumed that the coefficient of variation of study and auxiliary variables are equal. It is further assumed that under incomplete information, coefficient of variation of class is equal to coefficient of variation of population. Following this, we state thatTheorem 1.
TheMSE of the estimator P1υ to the fda is given by(6)MSEP1υ=Sy22λυ1−ρyz+θυ54−ρyz2.Theorem 2.
TheMSE of the estimator P2υ to the fda is given by(7)MSEP2υ=Sy2λυ1−ρyz2+θυ1−ρyz22.Remark 1.
TheMSE of the estimators P1υ and P2υ is given while considering the assumption made by Singh et al. [15, 16] that the population correlation coefficient is equal to the nonresponse class correlation coefficient ρyz2=ρyz:(8)MSEP1υ=Sy22λυ1−ρyz+θυ54−ρyz,MSEP2υ=Sy2λυ+θυ1−ρyz2.


### 4.2. Estimators Based on the Matched Portion of the Samplesm

It is assumed that there is no nonresponse on the first occasion as well on the matched portion of the sample. Under the above assumption, we consider the following estimators based on the matched samplesm of size m:(9)P1m=y¯m+byzmZ¯−z¯mexpx¯n−x¯mx¯n+x¯m,where byzm are the estimates of the population regression coefficients βyz, respectively, based on the sample of size m.To thefda, the MSE of the proposed estimators P1m is given by(10)MSEP1m=Y¯21m−1NCy21−ρyz2+1m−1n14Cx2−ρyxCyCx+ρyzρxzCyCx.Under the assumptionCx=Cz=Cy, the MSEs (10) reduce to(11)MSEP1m=Sy21m−1N1−ρyz2+1m−1n14−ρyx+ρyzρxz.


### 4.3. Covariance betweensu and sm

The covariance between (P1u, P1m) is given by(12)C11=CovP1u,P1m=−Sy2N1−ρyz2.It is to be noted that the expression (11) has been derived under the assumptions Cx=Cz=Cy, ρyz=ρyz2, and Cy=Cy2.(13)C21=CovP2u,P1m=−Sy2N1−ρyz2.


### 4.4. Linear Combination of Estimators

We have suggested the following estimators for estimating the population meanY¯ at the current occasion by combining the different estimators at the matched and unmatched portions of the samples, respectively:(14)P1=ϕ1P1u+1−ϕ1P1m,P2=ϕ2P2u+1−ϕ2P1m,where ϕi1≤ϕi≤1 is suitably chosen constant to be determined under certain assumptions.


### 4.5.MSEs of the Estimators P1 and P2

TheMSEs of the estimators are derived up to fda, and under the assumption, Cx=Cz=Cy, ρyz=ρyz2, and Cy=Cy2.Theorem 3.
MSEs of Pii=1, 2) to the fda are obtained as(15)MSEP1=ϕ12MSEP1u+1−ϕ12MSEP1m+2ϕ11−ϕ1C11,(16)MSEP2=ϕ22MSEP2u+1−ϕ22MSEP1m+2ϕ21−ϕ2C21.


### 4.6.MMSEs of the Estimators Pii=1, 2)

Since theMSE of the estimators Pii=1,2 in (15) and (16) is functions of unknown constants ϕi’s i=1, 2), therefore, the MSEs are minimized with respect to ϕii=1, 2), respectively, for(17)ϕ1opt=MSEP1υ−C11MSEP1υ+MSEP1m−2C11=1−μ1ℏ01−μ1fκ5−κ0ℏ01+μ1ℏ11+μ12ℏ21,(18)ϕ2opt=MSEP2υ−C21MSEP2υ+MSEP1m−2C21,=1−μ2ℏ02ℏ02+μ2ℏ12+μ12ℏ22.and thus, the resulting MMSEs of the estimators Pii=1, 2) are given as(19)MMSEP1=MSEP1υMSEP1m−C112MSEP1υ+MSEP1m−2C11=ƛ01+μ2ƛ11+μ22ƛ21ℏ01+μ2ℏ11+μ22ℏ21Sy2n,(20)MMSEP2=MSEP2υMSEP1m−C212MSEP2υ+MSEt1m−2C21=ƛ02+μ2ƛ12+μ22ƛ22ℏ02+μ2ℏ12+μ22ℏ22Sy2n.where κ0=21−ρyz, κ1=Θf2−15/4−ρyz, κ2=1/4−ρyx+ρxz,κ3=5/4−ρyz, κ4=1−ρyz2, κ5=1/4−ρyz+ρyzρxz, κ6=1+Θf2−1, ℏ01=κ0+κ1, ℏ11=1−fκ4−κ0−κ1, ℏ21=κ5−fκ4+fκ0, ℏ02=κ4κ6, ℏ12=κ41−κ6, ℏ22=κ5, ƛ01=1−fκ0+κ1κ4, ƛ11=κ0+κ1κ5+fκ1κ4−f2κ4κ4−κ0, ƛ21=f2κ4κ4−κ0−fκ0κ5, ƛ02=1−fκ42κ6, ƛ12=κ5κ6+fκ4κ6−1κ4, ƛ22=−fκ4κ5


### 4.7. Optimum Replacement Policy

It is observed from (18) and (19) that the MMSEs of the estimators Pii=1, 2) are the functions of μii=1, 2) (functions of sample to be drawn afresh at the second occasion); therefore, the optimum values of μi are obtained to estimate the population mean Y¯ with minimum precision and lowest cost. To obtain the optimum values of μii=1, 2), we minimize the MMSEs of the proposed estimators Pii=1, 2) given in (18) to (19), respectively, with respect to μii=1, 2) which result in quadratic equations in μi (i=1, 2), and the respective salutations of μii=1, 2) say μ^ii=1, 2) are given as follows:(21)ψ121μ12+2ψ021μ1+ψ011=0,(22)μ^1=−ψ021±ψ1212−ψ011ψ121D121,(23)ψ122μ22+2ψ022μ2+ψ012=0,(24)μ^2=−ψ022±ψ1222−ψ012ψ122ψ122,where(25)ψ011=ℏ01ƛ11−ƛ01ℏ11,ψ021=ℏ01ƛ21−ƛ01ℏ21,ψ121=ℏ11ƛ21−ƛ11ℏ21,ψ012=ℏ02ƛ12−ƛ02ℏ12,ψ022=ℏ02ƛ22−ƛ02ℏ22,ψ122=ℏ12ƛ22−ƛ12ℏ22.From (21) and (22), it is observed that real values of μii=1, 2) exist, iff the quantities under square roots are greater than or equals to zero, and for any combinations ρyx, ρyz, and ρxz, which satisfy the conditions of real situations, two real values of μii=1, 2) are possible, and hence, while choosing the values of μ^ii=1, 2), it should be remembered that 0≤μi≤1i=1, 2). All other values of μii=1, 2) are inadmissible. Substituting the admissible values of μ^i (say) μ^i0i=1, 2) from (21) and (22) in (17) and (18), respectively, we have the optimum values of mean squared errors of Pii=1, 2) which are shown as follows:(26)MMSEP10opt=ƛ01+μ10ƛ11+μ102ƛ21ℏ01+μ10ℏ11+μ102ℏ21Sy2n,MMSEP20opt=ƛ02+μ20ƛ12+μ202ƛ22ℏ02+μ20ℏ12+μ202ℏ22Sy2n.


### 4.8. Efficiency Comparisons

The percent relative losses in efficiencies of the estimatorsPii=1, 2) are obtained with respect to the similar estimator and natural successive sampling estimator when the nonresponse no observed on any occasion. The estimator Φ1is for complete information and under the similar assumption as estimator Pii=1, 2). Whereas the estimator Φ2 is natural estimator under successive sampling, given by(27)Φj=ψjΦju+1−ψjTjm;j=1,2,where Φ1u=y¯uexpZ¯−z¯u/Z¯+z¯u, Φ2u=y¯u,(28)T1m=P1m=y¯mZ¯z¯mexpx¯n−x¯mx¯n+x¯m,T2m=y¯m+byxmx¯n−x¯m.Proceeding in similar manner as discussed for the estimatorsPii=1, 2), the optimum mean squared errors of the estimators Φj, j=1, 2) are derived as(29)Φj=ψjΦju+1−ψjTjm,MMSEΦ10opt=A3+μ1∗0A2+μ1∗02A1B3+μ1∗0B2+μ1∗02B1Sy2n,MMSEΦ20opt=121+1−ρxy2−fSy2n,where μ1∗0=−Q2±Q22−Q1Q3/Q1 (fraction of the sample for the estimator Φ1),(30)Q1=B1A2−A1B2,Q2=B1A3−A1B3,Q3=B2A3−A2B3,A1=916f2κ02−f2κ0κ3−fκ2κ3,A2=fκ0κ3+κ2κ3−f1−fκ0κ3−916f2κ02,A3=1−fκ0κ3,B1=fκ0+κ2−32fκ0+fκ3,B2=1−fκ0−1+fκ3+32fκ0,B3=κ3.Remark 2.
For comparison of estimatorsPi and Φj, it was advisable by Cochran [3] & Feng and Zou [4] to assume the intraclass correlation coefficient equal, that is, ρxz=ρyz.


## 4.1. Estimators Based on the Unmatched Portion of the Amplesu

Looking the formulation of the estimatorsT1υ given by (1) due to Singh et al. [15], it is clear that the estimator T1υ is defined in the situation, in which information on the auxiliary variable z is obtained for all the sample units υ (drawn afresh from the entire population at the second occasion), and the population mean Z¯ of the auxiliary variable z is known, but some sample units fail to supply information on the study variable y. We note that, when suggesting the estimators for the population mean Y¯ on the current (second) occasion, Singh et al. [15] used only information on the sample mean z¯υ of the auxiliary variable z. However, one can also obtain the unbiased estimator z¯υ∗ of the population mean Z¯ (without any extra effort) while in the process of obtaining the unbiased estimator y¯υ∗ of the population mean Y¯. Thus, in this situation, where information on the auxiliary variable z is obtained for all the sample units υ, we have two unbiased estimators z¯υ∗ and z¯υ, of the population mean Z¯ of the auxiliary variable z (see, Singh and Kumar [17]). With this background, we suggest the following estimators (using z¯υ∗ and z¯υ together) for the population mean Y¯ on the current (second) occasion based on unmatched portion of the sample Sυ of the size υ.(4)P1υ=y¯υ∗expZ¯−z¯υZ¯+z¯υexpZ¯−z¯υ∗Z¯+z¯υ∗,P2υ=y¯υ∗exp2β^yzR^zZ¯−z¯υZ¯+z¯υexp4β^yz2R^zZ¯z¯υ−z¯υ∗Z¯+z¯υZ¯+z¯υ∗,where β^yz∗=syz∗/sz∗2,β^yz2∗=syz2∗/sz2∗2 are, respectively, the estimates of βyz2∗,βyz2∗,(5)syz∗=1υ−1∑sυ1yjzj+f∑sυ2hyjzj−υy¯υ∗z¯υ∗,sz∗2=1υ−1∑sυ1zj2+f∑sυ2hzj2−υz¯υ∗2,syz2∗=1υ2h−1∑sυ2hyj−y¯υ2hzj−z¯υ2h,sz2∗2=1υ2h−1∑sυ2hzj−z¯υ2h2,R^z=y¯υ∗Z¯,y¯υ2h=1υ2h∑sυ2hyj,z¯υ2h=1υ2h∑sυ2hzj.For the simplicity in calculation, Reddy (1974, [18]) and Ruiz Espejo [19] initially assumed that the coefficient of variation of study and auxiliary variables are equal. It is further assumed that under incomplete information, coefficient of variation of class is equal to coefficient of variation of population. Following this, we state thatTheorem 1.
TheMSE of the estimator P1υ to the fda is given by(6)MSEP1υ=Sy22λυ1−ρyz+θυ54−ρyz2.Theorem 2.
TheMSE of the estimator P2υ to the fda is given by(7)MSEP2υ=Sy2λυ1−ρyz2+θυ1−ρyz22.Remark 1.
TheMSE of the estimators P1υ and P2υ is given while considering the assumption made by Singh et al. [15, 16] that the population correlation coefficient is equal to the nonresponse class correlation coefficient ρyz2=ρyz:(8)MSEP1υ=Sy22λυ1−ρyz+θυ54−ρyz,MSEP2υ=Sy2λυ+θυ1−ρyz2.


## 4.2. Estimators Based on the Matched Portion of the Samplesm

It is assumed that there is no nonresponse on the first occasion as well on the matched portion of the sample. Under the above assumption, we consider the following estimators based on the matched samplesm of size m:(9)P1m=y¯m+byzmZ¯−z¯mexpx¯n−x¯mx¯n+x¯m,where byzm are the estimates of the population regression coefficients βyz, respectively, based on the sample of size m.To thefda, the MSE of the proposed estimators P1m is given by(10)MSEP1m=Y¯21m−1NCy21−ρyz2+1m−1n14Cx2−ρyxCyCx+ρyzρxzCyCx.Under the assumptionCx=Cz=Cy, the MSEs (10) reduce to(11)MSEP1m=Sy21m−1N1−ρyz2+1m−1n14−ρyx+ρyzρxz.


## 4.3. Covariance betweensu and sm

The covariance between (P1u, P1m) is given by(12)C11=CovP1u,P1m=−Sy2N1−ρyz2.It is to be noted that the expression (11) has been derived under the assumptions Cx=Cz=Cy, ρyz=ρyz2, and Cy=Cy2.(13)C21=CovP2u,P1m=−Sy2N1−ρyz2.


## 4.4. Linear Combination of Estimators

We have suggested the following estimators for estimating the population meanY¯ at the current occasion by combining the different estimators at the matched and unmatched portions of the samples, respectively:(14)P1=ϕ1P1u+1−ϕ1P1m,P2=ϕ2P2u+1−ϕ2P1m,where ϕi1≤ϕi≤1 is suitably chosen constant to be determined under certain assumptions.


## 4.5.MSEs of the Estimators P1 and P2

TheMSEs of the estimators are derived up to fda, and under the assumption, Cx=Cz=Cy, ρyz=ρyz2, and Cy=Cy2.Theorem 3.
MSEs of Pii=1, 2) to the fda are obtained as(15)MSEP1=ϕ12MSEP1u+1−ϕ12MSEP1m+2ϕ11−ϕ1C11,(16)MSEP2=ϕ22MSEP2u+1−ϕ22MSEP1m+2ϕ21−ϕ2C21.


## 4.6.MMSEs of the Estimators Pii=1, 2)

Since theMSE of the estimators Pii=1,2 in (15) and (16) is functions of unknown constants ϕi’s i=1, 2), therefore, the MSEs are minimized with respect to ϕii=1, 2), respectively, for(17)ϕ1opt=MSEP1υ−C11MSEP1υ+MSEP1m−2C11=1−μ1ℏ01−μ1fκ5−κ0ℏ01+μ1ℏ11+μ12ℏ21,(18)ϕ2opt=MSEP2υ−C21MSEP2υ+MSEP1m−2C21,=1−μ2ℏ02ℏ02+μ2ℏ12+μ12ℏ22.and thus, the resulting MMSEs of the estimators Pii=1, 2) are given as(19)MMSEP1=MSEP1υMSEP1m−C112MSEP1υ+MSEP1m−2C11=ƛ01+μ2ƛ11+μ22ƛ21ℏ01+μ2ℏ11+μ22ℏ21Sy2n,(20)MMSEP2=MSEP2υMSEP1m−C212MSEP2υ+MSEt1m−2C21=ƛ02+μ2ƛ12+μ22ƛ22ℏ02+μ2ℏ12+μ22ℏ22Sy2n.where κ0=21−ρyz, κ1=Θf2−15/4−ρyz, κ2=1/4−ρyx+ρxz,κ3=5/4−ρyz, κ4=1−ρyz2, κ5=1/4−ρyz+ρyzρxz, κ6=1+Θf2−1, ℏ01=κ0+κ1, ℏ11=1−fκ4−κ0−κ1, ℏ21=κ5−fκ4+fκ0, ℏ02=κ4κ6, ℏ12=κ41−κ6, ℏ22=κ5, ƛ01=1−fκ0+κ1κ4, ƛ11=κ0+κ1κ5+fκ1κ4−f2κ4κ4−κ0, ƛ21=f2κ4κ4−κ0−fκ0κ5, ƛ02=1−fκ42κ6, ƛ12=κ5κ6+fκ4κ6−1κ4, ƛ22=−fκ4κ5


## 4.7. Optimum Replacement Policy

It is observed from (18) and (19) that the MMSEs of the estimators Pii=1, 2) are the functions of μii=1, 2) (functions of sample to be drawn afresh at the second occasion); therefore, the optimum values of μi are obtained to estimate the population mean Y¯ with minimum precision and lowest cost. To obtain the optimum values of μii=1, 2), we minimize the MMSEs of the proposed estimators Pii=1, 2) given in (18) to (19), respectively, with respect to μii=1, 2) which result in quadratic equations in μi (i=1, 2), and the respective salutations of μii=1, 2) say μ^ii=1, 2) are given as follows:(21)ψ121μ12+2ψ021μ1+ψ011=0,(22)μ^1=−ψ021±ψ1212−ψ011ψ121D121,(23)ψ122μ22+2ψ022μ2+ψ012=0,(24)μ^2=−ψ022±ψ1222−ψ012ψ122ψ122,where(25)ψ011=ℏ01ƛ11−ƛ01ℏ11,ψ021=ℏ01ƛ21−ƛ01ℏ21,ψ121=ℏ11ƛ21−ƛ11ℏ21,ψ012=ℏ02ƛ12−ƛ02ℏ12,ψ022=ℏ02ƛ22−ƛ02ℏ22,ψ122=ℏ12ƛ22−ƛ12ℏ22.From (21) and (22), it is observed that real values of μii=1, 2) exist, iff the quantities under square roots are greater than or equals to zero, and for any combinations ρyx, ρyz, and ρxz, which satisfy the conditions of real situations, two real values of μii=1, 2) are possible, and hence, while choosing the values of μ^ii=1, 2), it should be remembered that 0≤μi≤1i=1, 2). All other values of μii=1, 2) are inadmissible. Substituting the admissible values of μ^i (say) μ^i0i=1, 2) from (21) and (22) in (17) and (18), respectively, we have the optimum values of mean squared errors of Pii=1, 2) which are shown as follows:(26)MMSEP10opt=ƛ01+μ10ƛ11+μ102ƛ21ℏ01+μ10ℏ11+μ102ℏ21Sy2n,MMSEP20opt=ƛ02+μ20ƛ12+μ202ƛ22ℏ02+μ20ℏ12+μ202ℏ22Sy2n.


## 4.8. Efficiency Comparisons

The percent relative losses in efficiencies of the estimatorsPii=1, 2) are obtained with respect to the similar estimator and natural successive sampling estimator when the nonresponse no observed on any occasion. The estimator Φ1is for complete information and under the similar assumption as estimator Pii=1, 2). Whereas the estimator Φ2 is natural estimator under successive sampling, given by(27)Φj=ψjΦju+1−ψjTjm;j=1,2,where Φ1u=y¯uexpZ¯−z¯u/Z¯+z¯u, Φ2u=y¯u,(28)T1m=P1m=y¯mZ¯z¯mexpx¯n−x¯mx¯n+x¯m,T2m=y¯m+byxmx¯n−x¯m.Proceeding in similar manner as discussed for the estimatorsPii=1, 2), the optimum mean squared errors of the estimators Φj, j=1, 2) are derived as(29)Φj=ψjΦju+1−ψjTjm,MMSEΦ10opt=A3+μ1∗0A2+μ1∗02A1B3+μ1∗0B2+μ1∗02B1Sy2n,MMSEΦ20opt=121+1−ρxy2−fSy2n,where μ1∗0=−Q2±Q22−Q1Q3/Q1 (fraction of the sample for the estimator Φ1),(30)Q1=B1A2−A1B2,Q2=B1A3−A1B3,Q3=B2A3−A2B3,A1=916f2κ02−f2κ0κ3−fκ2κ3,A2=fκ0κ3+κ2κ3−f1−fκ0κ3−916f2κ02,A3=1−fκ0κ3,B1=fκ0+κ2−32fκ0+fκ3,B2=1−fκ0−1+fκ3+32fκ0,B3=κ3.Remark 2.
For comparison of estimatorsPi and Φj, it was advisable by Cochran [3] & Feng and Zou [4] to assume the intraclass correlation coefficient equal, that is, ρxz=ρyz.


## 5. Numerical Illustrations

ForN=5000, n=500 and different choices of ρyx and ρyz, Tables 2 & 3 give the optimum values of μi0 and percent relative losses Lij, i=1, 2; j=1, 2) in the precision of estimators Pii=1, 2) with respect to Φj, j=1, 2).Table 1
Notations used and their meaning.
NotationsMeaningX¯,Y¯,Z¯Population mean of the study variabley and two auxiliary variables x, zy¯m,y¯υ,y¯υ1,y¯υ2h,y¯n,y¯n1,y¯n2hSample means of the study variable under different situationsx¯n,x¯mSample mean of auxiliary variable xz¯m,z¯υ,z¯υ1,z¯υ2hSample mean of auxiliary variable zρyx,ρxz,ρyzPopulation correlation coefficientsρ2yzPopulation correlation coefficient under nonresponseSx2,Sy2,Sz2Population variances of the study and auxiliary variablesS2y2,S2z2Population variances ofy and z under nonresponseΘ=N2/NProportion of nonresponding population to the total population at second occasiony¯υ∗=υ1y¯υ1+u2y¯υ2h/υHansen and Hurwitz estimator for study variableyz¯υ∗=υ1y¯υ1+u2y¯υ2h/υHansen and Hurwitz estimator for study variablex for the unmatched sample on the current occasionf2=υ2/υ2hCorrection factor under nonresponseTable 2
Percent relative lossesL11 and L12 with respect to Φ1 and Φ2 for f=0.1.
Θρyxρyzf2=1.5f2=2.0μ10L11L12μ10L11L120.100.500.800.78−32.63−119.120.90−26.42−108.850.850.63−49.64−176.930.72−42.89−164.440.900.53−79.78−277.200.60−70.70−258.160.950.43−152.55−510.500.49−134.20−466.160.600.800.78−28.02−110.430.90−22.03−100.580.850.63−44.13−165.960.72−37.62−153.960.900.53−72.77−262.250.60−64.05−243.960.950.43−142.10−486.310.49−124.52−443.730.700.800.78−23.16−99.140.90−17.39−89.810.850.63−38.33−151.690.72−32.09−140.330.900.53−65.44−242.810.60−57.09−225.510.950.43−131.22−454.850.49−114.42−414.550.800.800.78−18.01−84.130.90−12.48−75.510.850.63−32.26−132.710.72−26.30−122.220.900.53−57.83−216.970.60−49.86−200.970.950.43−119.99−413.020.49−104.01−375.760.150.500.800.84−29.55−114.031.02−20.03−98.300.850.67−46.26−170.690.80−36.15−151.980.900.56−75.19−267.580.66−62.02−239.960.950.46−143.08−487.610.55−118.03−427.070.600.800.84−25.05−105.551.02−15.87−90.450.850.67−40.88−159.960.80−31.14−142.000.900.56−68.36−253.010.66−55.71−226.480.950.46−133.02−464.330.55−109.01−406.180.700.800.84−20.30−94.521.02−11.46−80.230.850.67−35.22−146.010.80−25.87−129.010.900.56−61.22−234.070.66−49.10−208.960.950.46−122.55−434.040.55−99.62−379.020.800.800.84−15.27−79.861.02−6.80−66.640.850.67−29.28−127.470.80−20.35−111.750.900.56−53.80−208.890.66−42.24−185.670.950.46−111.74−393.790.55−89.92−342.910.200.500.800.90−26.42−108.851.14−13.56−87.610.850.72−42.89−164.440.89−29.51−139.680.900.60−70.70−258.160.73−53.76−222.620.950.49−134.20−466.160.60−103.66−392.320.600.800.90−22.03−100.581.14−9.62−80.180.850.72−37.62−153.960.89−24.74−130.180.900.60−64.05−243.960.73−47.77−209.840.950.49−124.52−443.730.60−95.23−372.810.700.800.90−17.39−89.811.14−5.46−70.510.850.72−32.09−140.330.89−19.72−117.830.900.60−57.09−225.510.73−41.50−193.210.950.49−114.42−414.550.60−86.46−347.440.800.800.90−12.48−75.511.14−1.04−57.660.850.72−26.30−122.220.89−14.47−101.410.900.60−49.86−200.970.73−34.99−171.110.950.49−104.01−375.760.60−77.40−313.71Table 3
Percent relative lossesL21 and L22 with respect to Φ1 and Φ2 for f=0.1.
Θρyxρyzf2=1.5f2=2.0μ20L21L22μ20L21L220.100.500.800.58−41.77−134.220.69−37.11−126.520.850.52−57.61−191.680.59−52.94−183.040.900.46−88.47−295.430.50−83.28−284.560.950.38−169.46−551.390.40−162.49−534.540.600.800.58−36.85−124.940.69−32.35−117.550.850.52−51.80−180.120.59−47.30−171.820.900.46−81.12−279.760.50−76.14−269.320.950.38−158.31−525.570.40−151.63−509.390.700.800.58−31.65−112.870.69−27.33−105.870.850.52−45.70−165.090.59−41.38−157.240.900.46−73.43−259.380.50−68.66−249.500.950.38−146.70−492.000.40−140.32−476.690.800.800.58−26.15−96.820.69−22.00−90.350.850.52−39.31−145.110.59−35.18−137.840.900.46−65.46−232.290.50−60.91−223.150.950.38−134.72−447.370.40−128.65−433.220.150.500.800.64−39.48−130.440.80−32.17−118.360.850.55−55.28−187.380.65−48.20−174.270.900.48−85.86−289.970.54−78.20−273.880.950.39−165.93−542.870.42−155.82−518.420.600.800.64−34.64−121.310.80−27.58−109.710.850.55−49.56−175.990.65−42.74−163.400.900.48−78.62−274.520.54−71.25−259.070.950.39−154.93−517.390.42−145.24−493.920.700.800.64−29.53−109.430.80−22.74−98.460.850.55−43.55−161.180.65−37.00−149.270.900.48−71.04−254.420.54−63.98−239.800.950.39−143.47−484.260.42−134.22−462.050.800.800.64−24.11−93.640.80−17.60−83.500.850.55−37.25−141.490.65−30.99−130.480.900.48−63.17−227.700.54−56.44−214.180.950.39−131.65−440.220.42−122.84−419.680.200.500.800.69−37.11−126.520.90−27.04−109.890.850.59−52.94−183.040.71−43.44−165.460.900.50−83.28−284.560.58−73.22−263.440.950.40−162.49−534.540.44−149.45−503.010.600.800.69−32.35−117.550.90−22.63−101.570.850.59−47.30−171.820.71−38.15−154.930.900.50−76.14−269.320.58−66.47−249.040.950.40−151.63−509.390.44−139.13−479.110.700.800.69−27.33−105.870.90−17.98−90.750.850.59−41.38−157.240.71−32.60−141.250.900.50−68.66−249.500.58−59.40−230.310.950.40−140.32−476.690.44−128.38−448.030.800.800.69−22.00−90.350.90−13.04−76.370.850.59−35.18−137.840.71−26.78−123.070.900.50−60.91−223.150.58−52.07−205.410.950.40−128.65−433.220.44−117.29−406.72The percent relative losses in the precision of estimatorsPii=1, 2) with respect to Φj, j=1, 2), under their respective optimality conditions are given by(31)Lij=MMSEPi0opt−MMSEΦi0optMMSEPi0opt,i=1,2;j=1.The following inferences may draw from Tables2 and 3.The above Table2 depicts that(i)
Considering the constant value ofΘ, f2, and ρyz, we observe that values of μ10, L11, and L12 decrease as ρyx increases. In other words, the intraclass correlation coefficient between y and z is inversely proportional to the new sample. That means the efficacy of the estimators under nonresponse depends positively on the auxiliary information. This attitude of the estimators is very important and extremely desirable.(ii)
Considering the constant value ofΘ, f2, and ρyz, we observe that values of μ20 remain the same whereas L11 and L12 increase as ρyx increases.(iii)
Considering the constant value ofΘ, ρyx, and ρyz, we observe that values of μ20, L11, and L12 increase as f2 increases.(iv)
Considering the constant value off2, ρyx, and ρyz, we observe that values of μ20, L11, and L12 increase as Θ increases. Table 3 depicts that nonresponse rate and sample size at current occasion are directly proportional and therefore the cost of the survey increases as well.(v)
Considering the constant value ofΘ, f2, and ρyz, we observe that values of μ20, L21, and L22 decrease as ρyx increases. In other words, the intraclass correlation coefficient between y and z is inversely proportional to the new sample. That means the efficacy of the estimators under nonresponse depends positively on the auxiliary information. This attitude of the estimators is very important and extremely desirable.(vi)
Considering the constant value ofΘ, f2, and ρyz, we observe that values of μ20 remain the same whereas L21 and L22 increase as ρyx increases.(vii)
Considering the constant value ofΘ, ρyx, and ρyz, we observe that values of μ20, L21, and L22 increase as f2 increases.(viii)
Considering the constant value off2, ρyx, and ρyz, we observe that values of μ20, L21, and L22 increase as Θ increases. In other words, the nonresponse rate and sample size at the current occasion are directly proportional; consequently, the cost of the survey increases as well.


## 6. Conclusions

In this paper, we have proposed exponential type estimators for estimating the population mean under the unavailability of full response in two-occasion successive sampling using the additional information. An extensive theoretical and computational study led to the conclusion that both the proposed estimators are better than the usual estimators available in the literature, and moreover, the estimatorP2 is better than the estimator P1 between the self-comparison. Therefore, the authors recommend the use of estimator P2 over the other estimators considered in this paper in practice. Furthermore, the researchers can use the composition of the proposed estimators for their use [20, 21].



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