Journal of Statistical Theory and Applications

In Press, Uncorrected Proof, Available Online: 21 May 2020

Generalized Chain Exponential-Type Estimators under Stratified Two-Phase Sampling with Subsampling the Nonrespondents

Authors
Aamir Sanaullah1, Muhammad Hanif2, *
1 COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan
2 National College of Business Administration and Economics, Lahore, Pakistan
*Corresponding author. Email: drmianhanif@gmail.com
Corresponding Author
Muhammad Hanif
Available Online 21 May 2020.
DOI
https://doi.org/10.2991/jsta.d.200507.002How to use a DOI?
Keywords
Auxiliary variables, Nonresponse, Stratified two-phase sampling, Exponential-type estimator, Chain estimator
Abstract

In this paper some generalized exponential-type chain estimators have been proposed for the finite population mean in the presence of nonresponse under stratified two-phase sampling when mean of another auxiliary variable is readily available. The expressions for the bias and mean square error of proposed estimators have been derived. The comparisons for proposed estimators have been made in theory with Hansen-Hurwitz’s, J. Am. Stat. Assoc. 41 (1946), 517–529, and Tabasum and Khan’s, J. Indian Soc. Agric. Stat. 58 (2004), 300–306, two-phase ratio and product estimators modified to the stratified sampling. An empirical study has also been carried out to demonstrate the performances of the estimators.

Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In a survey it is aimed to get hold of information on the subject of a target population. In prior of the survey the demarcation of the target population should be apparently affirmed. In an ideal world, all the selected units take part and make available, the requested information. On the other hand, the reality is not the same, notwithstanding how carefully the survey is planned and conducted, to acquire information on some of the units will not be possible, due to the variety of reasons even after callbacks, which is known as nonresponse.

In the presence of nonresponse, obtaining high response rates in the presence of nonresponse has been main aim of the survey statisticians. This growing interest is due to the significance of nonresponse bias in survey sampling. Madow et al. [1] discussed weighting adjustment and imputation methods to deal with 11 situations of nonresponse. Lessler and Kalsbeek [2] provided weighting adjustment and imputation procedure for 15 different situations. Rubin [3] suggest to ignore the incomplete information. This may be used where nonresponse is very low otherwise by doing such a method there occur a serious bias. Reweighting does not guarantee the adjustment of nonresponse bias. It may happen but most often one only can assume if the auxiliary information correlates strongly both with response propensity and study variable(s). If both of those conditions are satisfied the variance and mean square error (MSE) are reduced. A successful method of adjusting nonresponse bias is to use strongly correlated auxiliary information. In result of this, nonresponse and variance both may reduce Djerf [4,5], and Horngren [6]. Kalton ([7], p. 63) states “among the potential variables for use in forming weighting classes, the ones that are most effective in reducing nonresponse bias are those that are highly correlated both with the survey variables and the (0, 1) response variable.” Two types of auxiliary variables can be used if the auxiliary variables are known for all sampled units, then the adjustment is called sample-based; if they are known for the entire population, the adjustment is population-based [8,9]. The population-based adjustment is especially useful when the population totals are known. Sample-based adjustments need data for the full sample but do not require knowing control totals for the entire population. Sample- and population-based adjustments are equally effective for dealing with nonresponse bias [10,11]. Hansen and Hurwitz [12] were the first to develop a procedure to elicit response from the subsample of nonresponse. They envisaged an estimator for the estimation of population mean in the presence of nonresponse. Variance expression along with the optimum sampling fraction among nonrespondents was also derived. The procedure presented by Hansen-Hurwitz, is the edition of two-phase sampling, proposed by Neyman [13]. The technique was illustrated under simple random sampling design and it is also equally holds good for stratified sampling design and for other sampling designs.

Following Hansen-Hurwitz [12], Cochran [14] proposed a ratio estimator in simple random sampling for dealing with nonresponse. Okafor and Lee [15] advised a ratio estimator, which was first proposed by Khare and Srivastave [16] under two-phase sampling. Tabasum and Khan [17,18] extended the work done by Okafor and Lee [15] and studied some properties of the estimator in the presence of nonresponse under two-phase sampling. Singh et al. [19] developed some generalized exponential-type estimators under two-phase sampling to deal with response. Ismail et al. [20], Gamrot [21] and Shabbir and Khan [22] have recommended some improvements for the estimation of population mean in the presence of nonresponse using single or more auxiliary variables. Sanaullah et al. (2014b) proposed some generalized exponential-type estimators under stratified sampling for estimating population mean in two different situations of nonresponse.

1.1. Notations and Stratified Two-Phase Sampling with Subsampling the Nonrespondents

In many situations of practical importance, the population mean of either of the auxiliary variable, e.g. X¯h is not available in prior of a survey, in such a situation it is very usual to estimate it by the sample mean x¯h based on a preliminary first-phase sample of size nhn=h=1Lnh of which nhn=h=1Lnh is a subsample, i.e. nhnh. At the most, we use only knowledge of the population mean of another auxiliary variable, e.g. Z¯h, which is closely related to X¯h but remotely correlated to the main variable. That is if Z¯h is known to us, then it is advisable to estimate X¯h by X¯^h=x¯hZ¯z¯st, where h = 1, 2, …, L, which would provide a better estimate of X¯h than x¯h (Sanaullah et al., 2014a). Let us assume that at the first phase, all the nh units provide information on auxiliary characteristics. At the second phase from the sample nh, let nh(1) units provide the response for the requested information and nh(2) units do not. Following Hansen-Hurwitz [12] sub-sampling, a sub-sample of size rh from nh(2) non-respondents is selected at random and is approached for their direct interview such that rh=nh(2)/kh,kh>1. Here it is assumed that all the rh units provide the requested information.

When there occurs nonresponse on study variable as well as on the auxiliary variable, the usual two-phase ratio and product estimators for population mean are defined in stratified sampling respectively as

t1=y¯stx¯st/x¯st,Ratio estimator(1)
t2=y¯stx¯st/x¯st,Product estimator(2)
where y¯st and x¯st are Hansen-Hurwitz estimators modified to the stratified sampling for population means X¯ and Y¯ respectively and these are defined as y¯st=h=1LPhnh(1)y¯h(1)+nh(2)y¯h(2)r/nh, and x¯st=h=1LPhnh(1)x¯h(1)+nh(2)x¯h(2)r/nh with Ph=N/Nh, y¯h(1),x¯h(1), and y¯h(2)r,x¯h(2)r are the sample means for hth stratum based on nh(1) and nh2r units respectively, and x¯st=h=1LPhx¯h is the sample mean based on nh=h=1Lnh. It is to be pointed out that usual two-phase ratio estimator was Tabasum and Khan [17] in simple random sampling and t1 is modified form of Tabasum and Khan [17] to two-phase the stratified sampling. The MSEs for the ratio estimator t1 and product estimator t2 are given respectively as
MSE(t1)=h=1LPh2λhSyh2+λhSyh2+Rh2Sxh22RhSxyh+λhSyh(2)2+Rh2Sxh(2)22RhSxyh(2)(3)
and
MSE(t2)=h=1LPh2λhSyh2+λhSyh2+Rh2Sxh2+2RhSxyh+λhSyh(2)2+Rh2Sxh(2)2+2RhSxyh(2)(4)
where Syh2,Sxh2, and Syh(2)2,Sxh(2)2 are the variances from respondents and nonrespondents respectively with Rh=Y¯h/X¯h, λh=1nh1Nh, λh=1nh1Nh, λh=kh1nhWh(2) and Wh(2)=N2h/N.

When population means of the two auxiliary variables are available, Sanaullah et al. [23] proposed some exponential-type ratio-cum-ratio estimators for stratified two-phase sampling in the presence of nonresponse as

t3=h=1lPhy¯hexph=1lPhX¯hx¯hh=1lPhX¯h+a1x¯hexph=1lPhZ¯hz¯hh=1lPhZ¯h+(b1)z¯hExponential-type ratio-cum-ratio estimator(5)
where a,b are suitably chosen constants to be determined such that MSE of t3 is minimum.

The MSE of t3 is as

MSEt3Y¯2h=1lPh21Y¯2λhSyh2+λhSyh(2)2+1a2X¯2λhSxh2+1b2Z¯2λhSzh2+λhSzh(2)221aY¯X¯λhSyxh+1bZ¯Y¯λhSyzh+λhSyzh2λhSxzhabX¯Z¯(6)
Sanaullah et al. [24] envisaged an exponential-type chain ratio estimator under stratified two-phase sampling and population mean of an auxiliary variable x in not known but population mean of another variable z is on hand.

t4=h=1LPhy¯hexph=1LPhx¯hZ¯h=1LPhz¯hx¯hh=1LPhx¯hZ¯h=1LPhz¯h+x¯hExponential-type chain ratio(7)
The MSE of t4 is as
MSE(t4)Y¯2h=1LPh21Y¯2λhSyh2+141X¯2λhSxh2λhSxh2+1Z¯2λhSzh21Y¯X¯λhSxyhλhSxyh+1Y¯Z¯λhSyzh(8)
In this study, an attempt has been made for the development of generalized exponential-type chain ratio and product estimators using two auxiliary variables under stratified two-phase random sampling. The estimators have been proposed for the case when there occurs nonresponse on all the variables in second phase.

2. PROPOSED GENERALIZED EXPONENTIAL-TYPE CHAIN RATIO-CUM-RATIO AND PRODUCT-CUM-PRODUCT ESTIMATORS

Now it is assumed that information on a secondary auxiliary variable z is to be had. Then taking motivation from Sanaullah et al. [23,24], the inspiration of exponential-type chain ratio and exponential ratio-cum-ratio estimators have been combined together under stratified two-phase sampling design when there are auxiliary variables x and z which are correlated with study variable y in case of nonresponse. By following the same lines, another estimator (exponential-type chain product-cum-product estimator) has been proposed with its properties in the presence of nonresponse.

2.1. Generalized Exponential-Type Chain Ratio-Cum-Ratio Estimator

Motivated from Sanaullah et al. [23,24], we consider a form of an exponential-type chain ratio-cum-ratio estimator for stratified two-phase sampling in the presence of non-response as

tr(2.2)1=h=1LPhy¯hexp12h=1LPhx¯hh=1LPhx¯hZ¯h=1LPhz¯h+x¯hexp12h=1LPhz¯hh=1LPhZ¯h+z¯h,(9)
The estimator tr(2.2)1 in (9) leads to the form of generalized exponential-type chain ratio-cum-ratio estimator for population mean under stratified two-phase sampling in case of nonresponse as
tr(a,b)g=h=1LPhy¯hexp1ah=1LPhx¯hh=1LPhx¯hZ¯h=1LPhz¯h+(a1)h=1LPhx¯hexp1bh=1LPhz¯hh=1LPhZ¯h+(b1)h=1LPhz¯h(10)
where a0 and b0 are assumed unknown constants to be determined in such way whose values make the MSE of tr(a,b)g minimum.

It is observed that for various values of a and b in (10), we get various exponential-type chain ratio-cum-ratio estimators as deduced. From this class some examples are presented in Table 1 as follows:

Exponential-Type Chain Ratio-Cum-Ratio Estimators tr(a,b)g a b
tr(2,2)1=h=1LPhy¯hexp12h=1LPhx¯hh=1LPhx¯hZ¯h=1LPhz¯h+x¯hexp12h=1LPhz¯hh=1LPhZ¯h+z¯h 2 2
tr(2,1)2=h=1LPhy¯hexp12h=1LPhx¯hh=1LPhx¯hZ¯h=1LPhz¯h+x¯hexp1h=1LPhz¯hh=1LPhZ¯h` 2 1
tr(1,2)3=h=1LPhy¯hexp1h=1LPhx¯hh=1LPhx¯hZ¯h=1LPhz¯hexp12h=1LPhz¯hh=1LPhZ¯h+z¯h 1 2
Table 1

Some members of the class of the estimator tr(a,b)g.

In order to obtain the bias and mean square of the estimators, let us define

y¯h=Y¯h(1+e0h),x¯h=X¯h(1+e1h),x¯h=X¯h(1+e1h),z¯h=Z¯h(1+e2h),z¯h=Z¯h(1+e2h)ϑ200=V200V200,ϑ110=V110V110,Syh2=i=1NhyiY¯2Nh1,Syh(2)2=i=1Nh(2)yiY¯h(2)2Nh(2)1,(11)
where eih shows sampling error at second phase sampling in the presence of non-response, and eih shows sampling error at first phase sampling without nonresponse and we consider that E(eih)=E(eih)=0 where i = 0, 1, 2.

Let Vr,  s,  t=h=1LPhr+s+tEx¯hX¯hX¯ry¯hY¯hY¯sz¯hZ¯hZ¯t where r,s,t=0,1,2, and using (11), expectations are defined as

Ee02=1Y¯2h=1lPh2λhSyh2+λhSyh22=V020Ee12=1X¯2h=1lPh2λhSxh2=V200Ee12=1X¯2h=1lPh2λhSxh2+λhSxh22=V200Ee22=1Z¯2h=1lPh2λhSzh2+λhSzh22=V002Ee0.e2=1Y¯Z¯h=1lPh2λhSyzh+λhSyzh2=V011       Ee0.e1=1Y¯X¯h=1lPh2λhSxyh+λhSxyh2=V110Ee0.e1=1Y¯X¯h=1lPh2λhSyxh=V110Ee1.e2=1Z¯X¯h=1lPh2λhSxzh=V101,(12)
Using (11), the estimator in (10) can be expressed in the form of e’s as
tr(a,b)g=h=1LPhY¯1+e0hexph=1LPhX¯h1+e1hh=1LPhZ¯h1+e2hZ¯X¯h1+e1hh=1LPhX¯h1+e1hh=1LPhZ¯h1+e2hZ¯+(a1)X¯h1+e1hexph=1LPhZ¯hZ¯h1+e2hh=1LPhZ¯h+(b1)Z¯h1+e2h(13)
We expand the right-hand side of (13) and neglect the terms in ei higher than two. After some simplification we will have,
tr(a,b)gY¯Y¯e0+e1e1e2ae2b+e222a+e12a+(b1)e22b2a2e12+e12+e22b1b2e22+2b1b2e221ae1e1e1e2+e1e2e0e1+e0e1+e0e2+4a2e1e1e1e2+e1e2e0e2b1abe0e1e0e1+e0e2(14)
Using (14), the expressions for the bias and MSE of tr(a,b)g are obtained respectively as
Bias tr(a,b)gY¯1aϑ200+V0022+ϑ110+V011+3b1b2V0022a2ϑ200+V002V011b1abϑ110+V011(15)
and
MSEtr(a,b)gY¯2V020+1a2ϑ200+V002+V002b22aϑ110+V0112V011b+2abϑ101+V002(16)
The optimal values of a and b for which the MSE tr(a,b)g is minimum, are obtained as
aopt=BV002A2CV002AV011andbopt=BV002A2BV011ACwhereA=ϑ101+V002,  B=ϑ200+V002C=ϑ110+V011(17)
The minimum value of MSEtr(a,b)g as
min.MSEtr(aopt,bopt)gY¯2V020BV0112+C2V0022ACV011BV002A2(18)
The bias and MSE expressions for the class of estimators presented in Table 1, can be obtained by putting different values of a and b into (15) and (16) respectively, such as

For a=2 and b=2, the bias and MSE of tr(2,2)1 is obtained as

Biastr(2,2)1Y¯2ϑ002+V0022V20032ϑ110+V011V011(19)
and
MSEtr(2,2)1Y¯2V020+14ϑ200+V002+14V002ϑ110+V011V011+12ϑ101+V002(20)

For a=2 and b=1, the bias and MSE of tr(2,1)2 is obtained as

Bias tr(2,1)2Y¯V0024+V011(21)
and
MSEtr(2,1)2Y¯2V020+14ϑ200+V002+V002ϑ110+V0112V011+ϑ101+V002(22)

For a=1 and b=2, the bias and MSE of tr(1,2)3 is obtained as

Bias tr(1,2)3Y¯ϑ200+V0022+ϑ110+V011+34V0022ϑ200+V002V011212ϑ110+V011(23)
and
MSEtr(1,2)3Y¯2V020+ϑ200+V002+V00242ϑ110+V011V011+ϑ101+V002(24)

2.2. Generalized Exponential-Type Chain Product-cum-Product Estimator

Motivated from Sanaullah et al. (2014a, 2014b), we consider a form of an exponential-type chain product-cum-product estimator for stratified two-phase sampling in the presence of non-response as

tp(c,d)g=h=1LPhy¯hexpch=1LPhx¯hh=1LPhx¯hZ¯h=1LPhz¯h+(c1)h=1LPhx¯h1expdh=1LPhz¯hh=1LPhZ¯h+(d1)h=1LPhz¯h1,(25)
where c0 and d0 are assumed unknown constants to be determined in such way whose values make the MSE of tp(c,d)g minimum.

It is observed that for various values of c and d in (25), we get various exponential chain product-type estimators as deduced class of tp(c,d)g. From this class some examples can be considered in Table 2 as follows:

Exponential-Type Chain Product-Cum-Product Estimators tp(c,d)g c d
tp(2,2)1=h=1LPhy¯hexp2h=1LPhx¯hh=1LPhx¯hx¯hZ¯h=1LPhz¯h1exph=1LPhz¯hh=1LPhz¯hZ¯h1 2 2
tp(2,1)2=h=1LPhy¯hexph=1LPhx¯hh=1LPhx¯h+x¯hZ¯h=1LPhz¯h1exph=1LPhz¯hh=1LPhZ¯h1 2 1
tp(1,2)3=h=1LPhy¯hexph=1LPhx¯hh=1LPhx¯hZ¯h=1LPhz¯h1exph=1LPhz¯hh=1LPhz¯h+Z¯h1 1 2
Table 2

Some members of the class of the estimator tp(c,d)g.

We adapt the procedure (11)–(18), expressions for the bias and MSE of tp(c,d)g are obtained respectively as follows:

Biastp(c,d)g=Y¯1cϑ200+V0022+ϑ110+V011+d1d1d1d+2dV0022c2ϑ200+V002V011d1cdϑ110+V011(26)
and
MSEtp(c,d)gY¯2V020+1c2ϑ200+ϑ002+V002d2+2cϑ110+ϑ011+2V011d+2cdϑ101+ϑ002(27)
The optimum values of c and d are obtained as
copt=BV002A2CV002AV011    and    dopt=BV002A2BV011AC(28)
The minimum value of MSE(tp(c,d)g) is obtained as
min.MSEtp(copt,dopt)gY¯2V020BV0112+C2V0022ACV011BCV002A2(29)

The bias and MSE expressions for the estimators presented in Table 2 can be obtained directly from (26) and (27) respectively by putting different values of c and d.

3. EFFICIENCY COMPARISONS

Now we compare the proposed generalized exponential-type chain estimators with usual Hansen and Hurwitz’s [12] unbiased estimator y¯st and Tabasum and Khan [17] estimators t1 as

  1. Exponential-Type Chain Ratio-Cum-Ratio Estimators

    min.MSEtr(aopt,bopt)g<MSEy¯stif2V101V011V1104V1102V002+V200V0112<1    and    min.MSEtr(aopt,bopt)g<MSE(t1)ifϑ2002ϑ110V1012V200V002V002V1102+V200V01122V101V110V011<1(30)

  2. Exponential-Type Chain Product-Cum-Product Estimators

    MSEtp(copt,dopt)g<MSEy¯stif2V101V011V1104V1102V002+V200V0112<1    and    MSEtp(copt,dopt)g<MSE(t2)ifϑ200+2ϑ110V1012V200V002V002V1102+V200V01122V101V110V011<1(31)
    The proposed estimator will perform better if the above conditions hold.

4. EMPIRICAL RESULTS AND DISCUSSION

In order to examine the performance of proposed estimators under stratified two-phase sampling, we have taken two different stratified populations as,

Population-I: (Source: Koyuncu and Kadilar [25])

We consider number of teachers as study variable (Y), number of students as auxiliary variable (X) and number of classes in primary and secondary schools as another auxiliary variable (Z) for 923 districts at six 6 regions (1: Marmara, 2: Agean, 3: Mediterranean, 4: Central Anatolia, 5: Black Sea, and 6: East and Southeast Anatolia) in Turkey in 2007.

Population-II: [Source: Detailed livelihood assessment of flood affected districts of Pakistan September 2011, Food Security Cluster, Pakistan]

We consider food expenditure as study variable (Y), household earn as auxiliary variable (X) and total expenditure in May (2011) as another auxiliary variable (Z) for (6940) male and (1678) female households in flood affected districts of Pakistan September 2011.

The summery statistics for two populations are given in Appendix Table A.3. Form Table A.3 it is clear that correlations between study variable (Y) and auxiliary variables (X) and (Z) respectively ρxyh and ρyzh are positive in each stratum for population-I and these correlations are negative for population-II. It is therefore in order to examine the efficiency of chain ratio-cum-ratio estimators, population-I will be used and population-II is suitable for chain product-cum-product estimators to test empirically for their efficiency. We used stratified sampling for the selection of sample and Neyman allocation was used for allocating the sample size to different strata.

The comparison of proposed generalized exponential-type chain ratio-cum-ratio and exponential-type chain product-cum-product estimators with respect to Hansen and Hurwitz’s [12] have been made with Tabasum and Khan [17] modified to the stratified two-phase ratio and stratified two-phase product estimators respectively.

Table A.2 indicates MSE values of each estimator at three different nonresponse rates Wh2(10%, 20% and 30%), taking for each nonresponse rate four different inverse sampling rates kh(2.0, 2.5, 3.0 and 3.50). The percent relative efficiency (PRE) values for each estimator are computed with respect to the modified form of Hansen-Hurwitz [12] estimator y¯st in Table A.2 as,

PRE=Vary¯stMSEti(a,b)g×100
where g = 1, 2, 3 i = 1, r, p and (a, b) = {(2, 2), (2, 1), (1, 2) (aopt,bopt)}.

From Table A.1 it is noticed that PRE values for the proposed exponential-type chain ratio-cum-ratio estimators tr(2,2)1, tr(2,1)2, tr(1,2)3 and tr(aopt,bopt)g increase as the non-response rate increases from 10% to 30%. Similarly at each nonresponse rate, these PRE values increase for each estimator as the inverse sampling rate increases. Further it is observed that the PRE values of the proposed exponential-type chain ratio-cum-ratio estimators remain higher than the PRE values of Tabasum and Khan [17] ratio estimator (t1) modified to the two-phase sampling. This shows the proposed exponential-type chain ratio-cum-ratio estimators perform more efficiently. Furthermore it is scrutinized that tr(aopt,bopt)g is the most efficient estimator and from its class of exponential-type chain ratio-cum-ratio estimators tr(2,2)1, and tr(2,1)2 are the more efficient estimators.

From Table A.1 it is observed that the empirical results can be expressed same for the proposed exponential-type chain product-cum-product estimators tp(2,2)1, tp(2,1)2, and tp(copt,dopt)g. The only estimator tp(1,2)3 losses its PRE values if the nonresponse rate increases from 10% to 30% and due to the reason tp(1,2)3 remain no more efficient.

5. CONCLUSION

From the empirical results and discussion, finally it is concluded that the performance of generalized exponential-type chain ratio-cum-ratio (tr(aopt,bopt)g tr(2,2)1, & tr(2,1)2) and chain product-cum-product estimators (tp(copt,dopt)g, tp(2,2)1, & tp(2,1)2) is better for these populations on the basis of PRE values, and therefore, the class of generalized exponential-type chain estimators should be preferred for their practical applications in case of nonresponse.

APPENDIX

Wh2 kh Population No y¯st t1 tr(2,2)1 tr(2,1)2 tr(1,2)3 tr(aopt,bopt)g t2 tp(2,2)1 tp(2,1)2 tp(1,2)3 tp(copt,dopt)g
10% 2.0 1 100 316.5725 1628.54 1033.76 573.39 2702.14 —- —- —- —- —-
2 100 —- —- —- —- —- 64.9538 143.48 104.93 50.25 187.61
2.5 1 100 343.7379 1639.70 1092.53 596.35 2808.62 —- —- —- —- —-
2 100 —- —- —- —- —- 61.8850 137.57 100.24 47.75 183.39
3.0 1 100 369.9346 1649.63 1146.42 616.82 2910.62 —- —- —- —- —-
2 100 —- —- —- —- —- 59.3792 132.68 96.39 45.72 179.94
3.5 1 100 395.2162 1657.12 1195.26 634.99 3002.54 —- —- —- —- —-
2 100 —- —- —- —- —- 57.2944 128.58 93.18 44.04 177.11

20% 2.0 1 100 364.5528 1782.12 1049.51 555.31 2938.03 —- —- —- —- —-
2 100 —- —- —- —- —- 65.6052 146.53 106.00 50.27 194.31
2.5 1 100 412.3668 1842.74 1102.43 567.12 3121.84 —- —- —- —- —-
2 100 —- —- —- —- —- 62.9599 141.99 101.99 47.99 192.63
3.0 1 100 457.3057 1890.61 1145.43 576.28 3278.03 —- —- —- —- —-
2 100 —- —- —- —- —- 60.8689 138.33 98.81 46.19 191.34
3.5 1 100 499.6209 1929.9 1181.44 583.63 3415.24 —- —- —- —- —-
2 100 —- —- —- —- —- 59.17469 135.32 96.22 44.76 190.32

30% 2.0 1 100 382.2853 1799.28 1053.66 562.84 2958.40 —- —- —- —- —-
2 100 —- —- —- —- —- 76.5228 155.81 117.11 58.95 195.27
2.5 1 100 436.9706 1860.79 1103.44 576.17 3135.01 —- —- —- —- —-
2 100 —- —- —- —- —- 77.3685 154.62 116.79 59.39 194.52
3.0 1 100 487.7213 1908.43 1142.93 586.26 3283.78 —- —- —- —- —-
2 100 —- —- —- —- —- 78.0201 153.72 116.54 59.73 194.17
3.5 1 100 534.9462 1946.26 1174.94 594.19 3409.55 —- —- —- —- —-
2 100 —- —- —- —- —- 78.5378 153.03 116.35 59.99 194.09

(—-) shows data is not applicable.

Table A.1

Percent relative efficiencies (PREs) of estimators with respect to y¯st for different values of kheach at different rate of nonresponse under case-I using two different populations.

Wh2 kh Population No y¯st t1 tr(2,2)1 tr(2,1)2 tr(1,2)3 tr(aopt,bopt)g t2 tp(2,2)1 tp(2,1)2 tp(1,2)3 tp(copt,dopt)g
10% 2.0 1 2144.00 677.254 131.65 207.39 373.91 79.34 —- —- —- —- —-
2 5.09881 —- —- —- —- —- 7.8499 3.5536 4.8592 10.1473 2.7177
2.5 1 2370.93 689.749 144.59 217.01 397.57 84.41 —- —- —- —- —-
2 5.40353 —- —- —- —- —- 8.7316 3.9278 5.3903 11.3159 2.9464
3.0 1 2597.86 702.248 157.47 226.60 421.16 89.25 —- —- —- —- —-
2 5.70825 —- —- —- —- —- 9.6132 4.3023 5.9219 12.4855 3.1722
3.5 1 2824.79 714.745 170.46 236.33 444.85 94.07 —- —- —- —- —-
2 6.01298 —- —- —- —- —- 10.4949 4.6763 6.4527 13.6534 3.3949

20% 2.0 1 2540.35 696.839 142.54 242.05 457.45 86.46 —- —- —- —- —-
2 5.43362 —- —- —- —- —- 8.2823 3.7081 5.1259 10.8084 2.7964
2.5 1 2965.45 719.129 160.92 268.99 522.89 94.99 —- —- —- —- —-
2 5.90575 —- —- —- —- —- 9.3802 4.1593 5.7902 12.3072 3.0658
3.0 1 3390.55 741.419 179.33 296.01 588.35 103.43 —- —- —- —- —-
2 6.37788 —- —- —- —- —- 10.4781 4.6106 6.4545 13.8060 3.3331
3.5 1 3815.66 763.711 197.71 322.96 653.77 111.724 —- —- —- —- —-
2 6.85001 —- —- —- —- —- 11.5759 5.0621 7.1191 15.3055 3.5992

30% 2.0 1 2703.11 707.092 150.23 256.54 480.26 91.37 —- —- —- —- —-
2 6.62876 —- —- —- —- —- 8.6625 4.2542 5.6600 11.2445 3.3946
2.5 1 3209.59 734.510 172.48 290.87 557.05 102.37 —- —- —- —- —-
2 7.69847 —- —- —- —- —- 9.9504 4.9788 6.5917 12.9614 3.9575
3.0 1 3716.08 761.927 194.719 325.13 633.85 113.16 —- —- —- —- —-
2 8.76816 —- —- —- —- —- 11.2383 5.7040 7.5240 14.6804 4.5158
3.5 1 4222.56 789.343 216.95 359.38 710.64 123.84 —- —- —- —- —-
2 9.83786 —- —- —- —- —- 12.5263 6.4287 8.4558 16.3981 5.0687

(—-) shows data is not applicable.

Table A.2

MSEs of the estimators for different values of kh each at different rate of nonresponse under case-I using two different populations.

Population-I
Population-II
Stratum (h) 1 2 3 4 5 6 1 2
Stratified Mean, SDs and Correlation Coefficients Nh 127 117 103 170 205 201 6940 1678
nh 31 21 29 38 22 39 750 181
nh 70 50 75 95 70 90 1874 453
Syh 883.84 644.92 1033.4 810.58 403.65 711.72 21.4256 22.1319
Sxh 30486.7 15180.77 27549.69 18218.93 8497.77 23094.14 16625.33 12861.40
Szh 555.58 365.46 612.95 458.03 260.85 397.05 19394.09 16143.74
Y¯h 703.74 413 573.17 424.66 267.03 393.84 47.9805 48.0556
X¯h 20804.59 9211.79 14309..30 9478.85 5569.95 12997.59 18746.55 14303.98
Z¯h 498.28 318.33 431.36 311.32 227.20 313.71 19124.75 14742.47
ρxyh 0.9360 0.996 0.994 0.983 0.989 0.965 −0.4777 −0.4406
ρxzh 0.9396 0.9696 0.9770 0.9640 0.9670 0.9960 0.9138 0.8035
ρyzh 0.9790 0.976 0.984 0.983 0.964 0.983 −0.4422 −0.3547

Wh= 10% Nonresponse Syh2 510.57 386.77 1872.88 1603.3 264.19 497.84 20.4752 21.7407
Sxh2 9446.93 9198.29 52429.99 34794.9 4972.56 12485.10 18121.44 15492.72
Szh2 303.92 278.51 960.71 821.29 190.85 287.99 22010.50 20204.85
ρxy2 0.9961 0.9975 0.9998 0.9741 0.995 0.9284 −0.4826 −0.5422
ρxz2 0.9901 0.9895 0.9964 0.9609 0.9865 0.9752 0.8566 0.7691
ρyz2 0.9931 0.9871 0.99716 0.9942 0.985 0.9647 −0.3922 −0.3181

Wh= 20% Nonresponse Syh2 396.77 406.15 1654.4 1333.35 335.83 903.91 20.7359 22.6272
Sxh2 7439.16 8880.46 45784.78 29219.3 6540.43 28411.44 16155.37 13887.44
Szh2 244.56 274.42 965.42 680.28 214.49 469.86 19251.39 17323.10
ρxy2 0.9954 0.9931 0.996 0.9761 0.9966 0.9869 −0.4870 −0.4880
ρxz2 0.9897 0.9884 0.9789 0.9629 0.982 0.9825 0.8845 0.8399
ρyz2 0.9898 0.9798 0.9846 0.994 0.9818 0.9874 −0.4293 −0.3304

Wh= 30% Nonresponse Syh2 500.26 356.95 1383.7 1193.47 289.41 825.24 21.4660 22.4381
Sxh2 14017.994 7812.00 38379.77 26090.6 5611.32 24571.95 16877.33 12852.95
Szh2 284.4409 247.6279 811.21 631.28 188.30 437.90 19985.52 16007.36
ρxy2 0.9639 0.9919 0.9955 0.9801 0.9961 0.9746 −0.4808 −0.4395
ρxz2 0.9107 0.9848 0.9771 0.9650 0.9794 0.9642 0.8939 0.8298
ρyz2 0.9739 0.9793 0.9839 0.9904 0.9799 0.9829 −0.4347 −0.2823
Table A.3

Data statistics.

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Journal
Journal of Statistical Theory and Applications
Publication Date
2020/05
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.d.200507.002How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Aamir Sanaullah
AU  - Muhammad Hanif
PY  - 2020
DA  - 2020/05
TI  - Generalized Chain Exponential-Type Estimators under Stratified Two-Phase Sampling with Subsampling the Nonrespondents
JO  - Journal of Statistical Theory and Applications
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200507.002
DO  - https://doi.org/10.2991/jsta.d.200507.002
ID  - Sanaullah2020
ER  -