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

Aamir Sanaullah; Muhammad Hanif

doi:10.2991/jsta.d.200507.002

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Volume 19, Issue 2, June 2020, Pages 185 - 195

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

Authors

Aamir Sanaullah¹, Muhammad Hanif²^{, *}

¹COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan

²National College of Business Administration and Economics, Lahore, Pakistan

^*Corresponding author. Email: drmianhanif@gmail.com

Corresponding Author

Muhammad Hanif

Received 3 March 2019, Accepted 14 November 2019, Available Online 21 May 2020.

DOI: 10.2991/jsta.d.200507.002 How 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. Little and 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 nh′n′=∑h=1Lnh′ of which nhn=∑h=1Lnh is a subsample, i.e. nh⊂nh′. 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¯h′Z¯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¯st∗x¯st′ / x¯st∗, Ratio estimator(1)

t2=y¯st∗x¯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=1LPh2λh′Syh2+λhSyh2+Rh2Sxh2−2RhSxyh+λh∗Syh(2)2+Rh2Sxh(2)2−2RhSxyh(2)(3)

and

MSE(t2)=∑h=1LPh2λh′Syh2+λhSyh2+Rh2Sxh2+2RhSxyh+λh∗Syh(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=1nh−1Nh, λh′=1nh′−1Nh, λh∗=kh−1nhWh(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¯h∗exp∑h=1lPhX¯h−x¯h′∑h=1lPhX¯h+a−1x¯h′exp∑h=1lPhZ¯h−z¯h∗∑h=1lPhZ¯h+(b−1)z¯h∗ Exponential-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

MSEt3≈Y¯2∑h=1lPh21Y¯2λhSyh2+λh∗Syh(2)2+1a2X¯2λh′Sxh2+1b2Z¯2λhSzh2+λh∗Szh(2)2 −21aY¯X¯λh′Syxh+1bZ¯Y¯λhSyzh+λh∗Syzh2−λh′SxzhabX¯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¯hexp∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′−x¯h∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′+x¯h Exponential-type chain ratio(7)

The MSE of t4 is as

MSE(t4)≈Y¯2∑h=1LPh21Y¯2λhSyh2+141X¯2λhSxh2−λh′Sxh2+1Z¯2λh′Szh2−1Y¯X¯λhSxyh−λh′Sxyh+1Y¯Z¯λh′Syzh(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¯h∗exp1−2∑h=1LPhx¯h∗∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′+x¯h∗exp1−2∑h=1LPhz¯h∗∑h=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¯h∗exp1−a∑h=1LPhx¯h∗∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′+(a−1)∑h=1LPhx¯h∗exp1−b∑h=1LPhz¯h∗∑h=1LPhZ¯h+(b−1)∑h=1LPhz¯h∗(10)

where a≠0 and b≠0 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¯h∗exp1−2∑h=1LPhx¯h∗∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′+x¯h∗exp1−2∑h=1LPhz¯h∗∑h=1LPhZ¯h+z¯h∗	2	2
tr(2,1)2=∑h=1LPhy¯h∗exp1−2∑h=1LPhx¯h∗∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′+x¯h∗exp1−∑h=1LPhz¯h∗∑h=1LPhZ¯h`	2	1
tr(1,2)3=∑h=1LPhy¯h∗exp1−∑h=1LPhx¯h∗∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′exp1−2∑h=1LPhz¯h∗∑h=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∗=V200∗−V200′,ϑ110∗=V110∗−V110′,Syh2=∑i=1Nhyi−Y¯2Nh−1,Syh(2)2=∑i=1Nh(2)yi−Y¯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¯h−X¯hX¯ry¯h−Y¯hY¯sz¯h−Z¯hZ¯t where r,s,t=0,1,2, and using (11), expectations are defined as

Ee0∗2=1Y¯2∑h=1lPh2λhSyh2+λh∗Syh22=V020∗Ee1′2=1X¯2∑h=1lPh2λh′Sxh2=V200′Ee1∗2=1X¯2∑h=1lPh2λhSxh2+λh∗Sxh22=V200∗Ee2∗2=1Z¯2∑h=1lPh2λhSzh2+λh∗Szh22=V002∗Ee0∗.e2∗=1Y¯Z¯∑h=1lPh2λhSyzh+λh∗Syzh2=V011 ∗Ee0∗.e1∗=1Y¯X¯∑h=1lPh2λhSxyh+λh∗Sxyh2=V110∗Ee0∗.e1′=1Y¯X¯∑h=1lPh2λh′Syxh=V110′Ee1′.e2∗=1Z¯X¯∑h=1lPh2λh′Sxzh=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+e0hexp∑h=1LPhX¯h1+e1h′∑h=1LPhZ¯h1+e2h′Z¯−X¯h1+e1h∗∑h=1LPhX¯h1+e1h′∑h=1LPhZ¯h1+e2h′Z¯+(a−1)X¯h1+e1h∗exp∑h=1LPhZ¯h−Z¯h1+e2h∗∑h=1LPhZ¯h+(b−1)Z¯h1+e2h∗(13)

We expand the right-hand side of (13) and neglect the terms in e_i higher than two. After some simplification we will have,

tr(a,b)g−Y¯≈Y¯e0∗+e1′−e1∗−e2′a−e2∗b+e′222a+e1∗2a+(b−1)e2∗2b−2a2e1∗2+e′12+e′22−b−1b2e2∗2+2b−1b2e2∗2−1ae1∗e1′−e1∗e2′+e1′e2′−e0∗e1′+e0∗e1∗+e0∗e2′+4a2e1∗e1′−e1∗e2′+e1′e2′−e0∗e2∗b−1abe0∗e1∗−e0∗e1′+e0∗e2′(14)

Using (14), the expressions for the bias and MSE of tr(a,b)g are obtained respectively as

Bias tr(a,b)g≈Y¯1aϑ200∗+V002′2+ϑ110∗+V011′+3b−1b2V002∗−2a2ϑ200∗+V002′−V011∗b−1abϑ110∗+V011′(15)

and

MSEtr(a,b)g≈Y¯2V020∗+1a2ϑ200∗+V002′+V002∗b2−2aϑ110∗+V011′−2V011∗b+2abϑ101∗+V002′(16)

The optimal values of a and b for which the MSE tr(a,b)g is minimum, are obtained as

aopt=BV002∗−A2CV002∗−AV011∗ andbopt=BV002∗−A2BV011∗−ACwhereA=ϑ101∗+V002′, B=ϑ200∗+V002′C=ϑ110∗+V011′(17)

The minimum value of MSEtr(a,b)g as

min.MSEtr(aopt,bopt)g≈Y¯2V020∗−BV011∗2+C2V002∗−2ACV011∗BV002∗−A2(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)1≈Y¯2ϑ002∗+V002′2−V200′−32ϑ110∗+V011′−V011∗(19)

and

MSEtr(2,2)1≈Y¯2V020∗+14ϑ200∗+V002′+14V002∗−ϑ110∗+V011′−V011∗+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)2≈−Y¯V002′4+V011∗(21)

and

MSEtr(2,1)2≈Y¯2V020∗+14ϑ200∗+V002′+V002∗−ϑ110∗+V011′−2V011∗+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)3≈Y¯ϑ200∗+V002′2+ϑ110∗+V011′+34V002∗−2ϑ200∗+V002′−V011∗2−12ϑ110∗+V011′(23)

and

MSEtr(1,2)3≈Y¯2V020∗+ϑ200∗+V002′+V002∗4−2ϑ110∗+V011′−V011∗+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¯h∗expc∑h=1LPhx¯h∗∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′+(c−1)∑h=1LPhx¯h∗−1expd∑h=1LPhz¯h∗∑h=1LPhZ¯h+(d−1)∑h=1LPhz¯h∗−1,(25)

where c≠0 and d≠0 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¯h∗exp2∑h=1LPhx¯h∗∑h=1LPhx¯h∗−x¯h′Z¯∑h=1LPhz¯h′−1exp∑h=1LPhz¯h∗∑h=1LPhz¯h∗−Z¯h−1	2	2
tp(2,1)2=∑h=1LPhy¯h∗exp∑h=1LPhx¯h∗∑h=1LPhx¯h∗+x¯h′Z¯∑h=1LPhz¯h′−1exp∑h=1LPhz¯h∗∑h=1LPhZ¯h−1	2	1
tp(1,2)3=∑h=1LPhy¯h∗exp∑h=1LPhx¯h∗∑h=1LPhx¯h′Z¯∑h=1LPhz¯h′−1exp∑h=1LPhz¯h∗∑h=1LPhz¯h∗+Z¯h−1	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∗+V002′2+ϑ110∗+V011′+d−1d1−d−1d+2dV002∗−2c2ϑ200∗+V002′−V011∗d−1cdϑ110∗+V011′(26)

and

MSEtp(c,d)g≈Y¯2V020∗+1c2ϑ200∗+ϑ002′+V002∗d2+2cϑ110∗+ϑ011′+2V011∗d+2cdϑ101∗+ϑ002′(27)

The optimum values of c and d are obtained as

copt=−BV002∗−A2CV002∗−AV011∗ and dopt=−BV002∗−A2BV011∗−AC(28)

The minimum value of MSE(tp(c,d)g) is obtained as

min.MSEtp(copt,dopt)g≈Y¯2V020∗−BV011∗2+C2V002∗−2ACV011∗BCV002∗−A2(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

Exponential-Type Chain Ratio-Cum-Ratio Estimators
min.MSEtr(aopt,bopt)g<MSEy¯st∗if2V101′V011∗V110′4V110′2V002∗+V200′V011∗2<1 and min.MSEtr(aopt,bopt)g<MSE(t1)ifϑ200∗−2ϑ110∗V101′2−V200′V002∗V002∗V110′2+V200′V011∗2−2V101′V110′V011∗<1(30)
Exponential-Type Chain Product-Cum-Product Estimators
MSEtp(copt,dopt)g<MSEy¯st∗if2V101′V011∗V110′4V110′2V002∗+V200′V011∗2<1 and MSEtp(copt,dopt)g<MSE(t2)ifϑ200∗+2ϑ110∗V101′2−V200′V002∗V002∗V110′2+V200′V011∗2−2V101′V110′V011∗<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 W_h2(10%, 20% and 30%), taking for each nonresponse rate four different inverse sampling rates k_h(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¯st∗MSEti(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.

CONFLICT OF INTEREST

There is no conflict of interest involved and the research was carried out with authors's own contribution without any outside funding.

ACKNOWLEDGMANTS

The authors thank the Associate Editor and anonymous reviewers for their useful comments and suggestions on an earlier version of this manuscript which led to this improved one.

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	—-	—-	—-	—-	—-
	2.0	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	—-	—-	—-	—-	—-
	2.5	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	—-	—-	—-	—-	—-
	3.0	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	—-	—-	—-	—-	—-
	3.5	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	—-	—-	—-	—-	—-
	2.0	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	—-	—-	—-	—-	—-
	2.5	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	—-	—-	—-	—-	—-
	3.0	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	—-	—-	—-	—-	—-
	3.5	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	—-	—-	—-	—-	—-
	2.0	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	—-	—-	—-	—-	—-
	2.5	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	—-	—-	—-	—-	—-
	3.0	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	—-	—-	—-	—-	—-
	3.5	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	—-	—-	—-	—-	—-
	2.0	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	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	—-	—-	—-	—-	—-
	3.0	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	—-	—-	—-	—-	—-
	3.5	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	—-	—-	—-	—-	—-
	2.0	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	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	—-	—-	—-	—-	—-
	3.0	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	—-	—-	—-	—-	—-
	3.5	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	—-	—-	—-	—-	—-
	2.0	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	—-	—-	—-	—-	—-
	2.5	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	—-	—-	—-	—-	—-
	3.0	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	—-	—-	—-	—-	—-
	3.5	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	N_h	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
	S_yh	883.84	644.92	1033.4	810.58	403.65	711.72	21.4256	22.1319
	S_xh	30486.7	15180.77	27549.69	18218.93	8497.77	23094.14	16625.33	12861.40
	S_zh	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

W_h= 10% Nonresponse	S_yh2	510.57	386.77	1872.88	1603.3	264.19	497.84	20.4752	21.7407
	S_xh2	9446.93	9198.29	52429.99	34794.9	4972.56	12485.10	18121.44	15492.72
	S_zh2	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

W_h= 20% Nonresponse	S_yh2	396.77	406.15	1654.4	1333.35	335.83	903.91	20.7359	22.6272
	S_xh2	7439.16	8880.46	45784.78	29219.3	6540.43	28411.44	16155.37	13887.44
	S_zh2	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

W_h= 30% Nonresponse	S_yh2	500.26	356.95	1383.7	1193.47	289.41	825.24	21.4660	22.4381
	S_xh2	14017.994	7812.00	38379.77	26090.6	5611.32	24571.95	16877.33	12852.95
	S_zh2	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
Volume-Issue: 19 - 2
Pages: 185 - 195
Publication Date: 2020/05/21
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TY  - JOUR
AU  - Aamir Sanaullah
AU  - Muhammad Hanif
PY  - 2020
DA  - 2020/05/21
TI  - Generalized Chain Exponential-Type Estimators under Stratified Two-Phase Sampling with Subsampling the Nonrespondents
JO  - Journal of Statistical Theory and Applications
SP  - 185
EP  - 195
VL  - 19
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200507.002
DO  - 10.2991/jsta.d.200507.002
ID  - Sanaullah2020
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