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Volume 18, Issue 1, March 2019, Pages 12 - 25

A Family of Ratio Estimators in Stratified Random Sampling Utilizing Auxiliary Attribute Along Side the Nonresponse Issue

Authors
Usman Shahzad1, *, Muhammad Hanif1, Nursel Koyuncu2, *, A. V. Garcia Luengo3
1Department of Mathematics and Statistics, PMAS-Arid Agriculture University, Rawalpindi, Pakistan
2Department of Statistics, University of Hacettepe, Ankara, Turkey
3Department of Mathematics, University of Almeria, Almeria, Spain
*Corresponding authors. Email: nkoyuncu@hacettepe.edu.tr; usman.stat@yahoo.com
Corresponding Authors
Usman Shahzad, Nursel Koyuncu
Received 7 April 2017, Accepted 28 March 2018, Available Online 22 April 2019.
DOI
10.2991/jsta.d.190306.002How to use a DOI?
Keywords
Nonresponse; stratified random sampling; attribute
Abstract

This article proposes some estimators based on an adaptation of the estimators developed by Bahl and Tuteja [1], Diana [2], Koyuncu and Kadilar [3], Koyuncu and Kadilar [4], Shabbir and Gupta [5], and Koyuncu [6] utilizing available supplementry attributes. Further, a new family of estimators is also developed by adapting Koyuncu [7] and Koyuncu [6]. Under stratified sampling scheme along side the nonresponse issue, the expressions for the the mean square errors (MSEs) of the adapted and proposed estimators have been determined. These theoretical findings are demonstrated by a numerical illustration with original data set.

Copyright
© 2019 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 survey sampling, utilization of the auxiliary information results in significant improvement in the accuracy/precision of estimators of the population mean. When auxiliary information is accessible, the analysts need to use it in the strategy for estimation to acquire the most productive estimator. Ratio method is utilized to get more efficient estimates for the population mean by utilizing the correlation between study and auxiliary variables. Hansen et al. [8] and Kaur [9] proposed some estimators utilizing auxiliary information under stratified random sampling. Diana [2] also developed a class of estimators of the population mean by examining their mean square errors (MSEs) upto the kth order approximation. After that Koyuncu and Kadilar [10] extended the idea of Diana [2]. Sisodia and Dwivedi [11] introduced a ratio estimator by using coefficient of variation of an auxiliary variable. Singh and Kakran [12] proposed another ratio estimator by using known coefficient of kurtosis of an auxiliary variable. Upadhyaya and Singh [13] also suggested a ratio type estimator by using coefficient of variation and kurtosis of an auxiliary variable. The estimators of Sisodia and Dwivedi [1], Singh and Kakran [12], and Upadhyaya and Singh [13] belong to simple random sampling scheme. Kadilar and Cingi [14] give the stratified versions of these abovementioned estimators. Kadilar and Cingi [15] and Koyuncu and Kadilar [4] developed some estimators by extending the idea of Prasad [16].

All of the above discussion shows that much of literature is available on the estimation of population mean under stratified random sampling using available auxiliary information. Recently, Koyuncu [7] introduced, a family of estimators under stratified random sampling scheme using known parameters of auxiliary attribute. So we extend the work of Koyuncu [7], by introducing the estimators of Bahl and Tuteja [1], Diana [2], Koyuncu and Kadilar [3], Koyuncu and Kadilar [4], Shabbir and Gupta [5] for the situation when auxiliary attribute is available. Note that these estimators were developed when auxiliary variable is known, but in some situations it is not possible to obtain information on auxiliary variable, in these situations we look for auxiliary character and, if available, use it in estimation stage. So we are introducing these estimators for the case when auxiliary character (attribute) is known. Also, we propose a new class of estimators in this filed by taking motivation from Koyuncu [7] and Koyuncu [6]. Further we also extend the work for nonresponse issue, when nonresponse present in variate of interest Y. For detail study see Chaudhary et al. [17].

Remaining part of this paper is composed as follow. In Sections 2 and 3, we quickly present preliminaries and reviewed estimators when auxiliary information is available. Adapted estimators are presented in Section 4. In Section 5, a new class of estimators is proposed for stratified random sampling along with their minimum MSE. Section 6, is devoted for nonresponse issue. Analytical examinations are appeared in Section 7 to represent that the proposed estimator are dominant than the existing ones. Finally, in order to check the performance of existing estimators with the proposed estimators, a numerical illustration has been carried out using real (natural) population in Section 8.

2. NOTATIONS

Suppose U=U1,U2,…,UN be the finite population containing N units. Let ϕ is an auxiliary character (attribute) and Y is the study variable taking values yhi and ϕhi, respectively, in the unit i=1,2,…,N in the hth stratum consisting of Nh units such that ∑h=1LNh=N. Let nh be the size of the sample drawn from the hth stratum by using simple random sampling without replacement scheme such that ∑h=1Lnh=n. Further, we consider that ϕh takes only binary values (0, 1) as

  • ϕi=1, if the ith unit of the population possesses attribute ϕ,

  • ϕi=0, otherwise.

Let A=∑i=1Nϕi, a=∑i=1nϕi, Ah=∑i=1Nhϕhi, ah=∑i=1nhϕhi denote the total number of units in the population, population stratum h and sample stratum h possessing an auxiliary attribute ϕ, respectively. The corresponding population and sample, h-th proportions are Ph=AhN and ph^=ahn, respectively.

Moreover y¯st=∑h=1LWhy¯h, where y¯h=1nh∑i=1nhyhi; Y¯=Y¯st=∑h=1LWhY¯h and Wh=NhN is the stratum weight. We can define similar expressions for p also. Further ψs=∑h=1LWhSϕh, ψc=∑h=1LWhCϕh, ψb=∑h=1LWhβ2ϕh and ψr=∑h=1LWhρϕh.

For finding MSE of the proposed and existing estimators, we use the following notation

eost=y¯st−Y¯Y¯,     Eeost2c=∑h=1LWh2fh′Syh2Y¯2=δ2.0,     Eeost=0,
e1st=pst−PP,     Ee1st2=∑h=1LWh2fh′Sxh2X¯2=δ0.2,     Ee1st=0,
f′=1nh−1Nh,     Eeoste1st=∑h=1LWh2fh′Sxyh2Y¯2=δ1.1
and
δa.b=∑h=1LWha+bE(yhi−Y¯h)a(phi−Ph)bY¯aPb.

Now, we describe some existing estimators with their MSEs in the next section.

3. EXISTING ESTIMATORS

The variance of the sample mean y¯st in stratified random sampling with out replacement is given by

Vy¯st=Y¯2δ2.0.

The stratified version of customary ratio estimator for mean is given by

y¯^Rst=y¯stpstP.

The MSE of customary ratio estimator given in Eq. (2) for the first order approximation is given below

MSEy¯^Rst=Y¯2δ2.0+δ0.2−2δ1.1.

The traditional Regression estimator is given by,

y¯^stlr=y¯st+bstP−pst.

The MSE of y¯^stlr is

MSEy¯^stlr=Y¯2δ2.01−ρst2,
where ρst=δ1.1δ2.0δ0.2.

Utilizing auxiliary attributes, Koyuncu [7] introduced the following family of estimators under stratified random sampling as follows:

y¯^kkst=wc1y¯st+wc2P−pstastP+bstastpst+bst,
where a≠0 and b be any known characteristics of auxiliary information.

Some family members of Koyuncu [7] are shown in Table 1.

y¯^kkst ast bst νj
y¯^kk1st 1 ψr ν3
y¯^kk2st 1 ψc ν2
y¯^kk3st ψr ψc ν4
y¯^kk4st ψc ψr ν5
Table 1

Some family members of Koyuncu [7].

MSEy¯^kkst=Y¯2+wc12Akk+wc22Bkk+2wc1wc2Ckk−2wc1Dkk−2wc2Ekk.

The minimum MSE of y¯^kkst for

wc1opt=BkkDkk−CkkEkkAkkBkk−Ckk2 and wc2opt=−CkkDkk+AkkEkkAkkBkk−Ckk2
is given by
MSEminy¯^kkst=Y¯2−BkkDkk2+AkkEkk2−2CkkDkkEkkAkkBkk−Ckk2,
where
Akk=Y¯21+δ2.0+3νj2δ0.2−4νjδ1.1,
Bkk=P2δ0.2,
Ckk=PY¯2νjδ0.2−δ1.1,
Dkk=Y¯21+νj2δ0.2−νjδ1.1,
Ekk=PY¯νjδ0.2.

4. ADAPTED ESTIMATORS

By adapting Bahl and Tuteja [1], we propose the following exponential estimator under stratified sampling using auxiliary attributes

y¯^BTst=y¯stexpP−pstP+pst.

The MSE of y¯^BTst for the first order approximation are given below:

MSEy¯^BTst=Y¯2δ2.0+14δ0.2−δ1.1.

By adapting Diana [2], we propose the following family of estimators under stratified sampling using auxiliary attributes

y¯^di=y¯stpstPm1wd+1−wdpstPm2m3,
where m1, m2, m3, and wd can take values finitely.

Koyuncu and Kadilar [3] constructed the generalized version of Diana [2]. By adapting Koyuncu and Kadilar [3], we constructed the generalized version of y¯^di, utilizing available auxiliary attributes as

y¯^kd=y¯stpstPm4wkd+1−wkdastpst+bstastP+bstm5m6, 
where m4, m5, m6, and wkd can take values finitely.

It is interesting to note that y¯^kd, y¯^di, and y¯^stlr have equal MSE.

By adapting Koyuncu and Kadilar [4], we developed the following ratio estimators under stratified sampling, utilizing auxiliary attributes as

y¯^kist=ky¯staP+bαastpst+bst+1−αastP+bstM fori=1,2,…,6,
where M=1 for ratio type estimators, and α=1. The MSE of y¯^kist is given by
MSEy¯^kist=Y¯2k2δ2.0+gα2νj2δ0.2−2Mανjδ1.12k2−k+(k−1)2,
where
g=k22M2+M−kM2+M.

The optimum value of k is

kopt=Ak2Bk.

Some family members of Koyuncu and Kadilar [4] are shown in Table 2.

y¯^k

νj
y¯^k1st ν1=PP+ψs
y¯^k2st ν2=PP+ψc
y¯^k3st ν3=PP+ψr
y¯^k4st ν4=ψrPψrP+ψc
y¯^k5st ν5=ψcPψcP+ψr
y¯^k6st ν6=ψrPψrP+ψs
Table 2

Some family members of Koyuncu and Kadilar [4].

The minimum mean squared error of y¯^kist is

MSEminy¯^kist=Y¯21−Ak24Bk,
where
Ak=M2+Mα2νj2δ2.0−2Mανjδ1.1+2,
Bk=δ0.2+2M2+Mα2νj2δ0.2−4Mανjδ1.1+1.

By adapting Shabbir and Gupta [5], we introduce an exponential ratio estimator as

y¯^sgst=wsg1y¯st+wsg2P−pstexpA¯−a¯stA¯+a¯st,
where
a¯=pst+NP,A¯=P+NP.

The MSE of y¯^sgst is

MSEy¯^sgst=Y¯2+wsg12Asg+wsg22Bsg+2wsg1wsg2Csg−2wsg1Dsg−wsg2Esg.

The minimum MSE of y¯^sgst for

wsg1opt=BsgDsg−CsgEsg2AsgBsg−Csg2 and wsg2opt=−CsgDsg+AsgEsg2AsgBsg−Csg2
is given by
MSEminy¯^sgst=Y¯2−BsgDsg2+AsgEsg24−CsgDsgEsgAsgBsg−Csg2,
where
Asg=Y¯21+δ2.0+δ0.2(1+N)2−2δ1.11+N,
Bsg=P2δ0.2,
Csg=PY¯δ0.21+N−δ1.1,
Dsg=Y¯21+3δ0.28(1+N)2−δ1.121+N,
Esg=PY¯δ0.21+N.

By adapting Koyuncu [6], we introduce exponential family of ratio estimators as

y¯^nkst=w1nky¯st+w2nkpstPγ′expUstP−pstUstP+pst−2Vst.

Some family members of Koyuncu [6] are shown in Table 3.

y¯^nk Ust Vst γ′
y¯^nk1st ψb ψc 1
y¯^nk2st ψb ψc 2
y¯^nk3st ψc ψb 1
y¯^nk4st ψc ψb 2
y¯^nk5st ψr ψc 1
y¯^nk6st ψr ψc 2
Table 3

Some family members of Koyuncu [6].

The MSE of y¯^nkst is

MSEy¯^nkst=Y¯2+w1nk2Ank+w2nk2Bnk+2w1nkw2nkCnk−2w1nkDnk−2w2nkEnk.

The minimum MSE of y¯^nkst for

w1nkopt=BnkDnk−CnkEnkAnkBnk−Cnk2 and w2nkopt=−CnkDnk+AnkEnkAnkBnk−Cnk2
is given by
MSEminy¯^nkst=Y¯2−BnkDnk2+AnkEnk2−2CnkDnkEnkAnkBnk−Cnk2.
where
Ank=Y¯21+δ2.0+θnk2δ0.2−2θnkδ1.1,
Bnk=1+ank2+2cnkδ0.2,
Cnk=Y¯1+38θnk2−12ankθnk+cnkδ0.2+ank−12θnkδ1.1,
Dnk=Y¯21−38θnk2δ0.2−12θnkδ1.1,
Enk=Y¯1+cnkδ0.2.

Further

ank=γ−12θnk, bnk=γγ−12, cnk=bnk−12γθnk+38θnk2, θnk=PUstPUst+Vst.

5. PROPOSED CLASS OF ESTIMATORS

Taking motivation from Koyuncu [7], we propose the following class of estimators as

y¯^Nst=ψ′1astP+bstβastpst+bst+1−βastP+bstg,
where
ψ′1=w1y¯st2Ppst+pstP+w2pstPγ.

We can write y¯^Nst as

y¯^Nst=w1Y¯1+eost−ae1st+b+12e1st2−aeoste1st+w21+γ−ae1st+ce1st2,
where
a=gβv′, b=gg+12β2v′2, c=b−aγ+γγ−12, and v′=astPastP+bst.

The bias of y¯^Nst is

By¯^Nst=w1Y¯1+b+12δ0.2−aδ11+w21+cδ0.2−Y¯.

The MSE of y¯^Nst is

MSEy¯^Nst=L1+w12ϕA1+w22ϕB1+2w1w2ϕC1−2w1ϕD1−w2ϕE1,
where
L1=Y¯2, ϕA1=Y¯21+δ2.0+a2+2b+1δ0.2−2aδ11, ϕB1=1+(γ−a)2+2cδ0.2,
ϕC1=Y¯1+c−aγ−a+b+12δ0.2+γ−2aδ1.1,
ϕD1=Y¯21+b+12δ0.2−aδ1.1,
ϕE1=2Y¯1+cδ1.1.

Some family members of proposed class are shown in Table 4.

y¯^Nst γ g ast bst β
y¯^N1st 12 1 ψb ψc 1
y¯^N2st 12 1 ψc ψb 1
y¯^N3st 12 1 1 ψr 1
y¯^N4st 12 1 1 ψb 1
y¯^N5st 12 1 1 ψc 1
y¯^N6st 13 1 ψc ψr 1
y¯^N7st 12 1 1 ψb 1
y¯^N8st 1 2 ψc ψr 1
y¯^N9st 1 2 ψr ψc 1
Table 4

Some family members of proposed class.

By partially differentiating MSE of y¯^Nst w.r.t w1 and w2, we get the optimum values of w1, w2, that is,

w1opt=ϕB1ϕD1−ϕC1ϕE12ϕA1ϕB1−ϕC12,
and
w2opt=−ϕC1ϕD1+ϕA1ϕE12ϕA1ϕB1−ϕC12.

By putting w1opt, w2opt in MSEy¯^Nst and get minimum MSE of y¯^Nst, that is,

MSEminy¯^Nst=Y¯2L−ANBN.
where AN=ϕBϕD2+ϕAϕE24−ϕCϕDϕE and BN=ϕAϕB−ϕC2.

6. NONRESPONSE

Hansen and Hurwitz [18] estimator in case of nonresponse under stratified sampling scheme for hth stratum is as follows:

y¯h′=nh1y¯h1+nh2y¯h2nh,
where nh1 is the sample mean of response group and nh2=nh22l is the mean of subsample of nonresponse group in the hth stratum.

Using Hansen and Hurwitz [18] subsampling scheme, an unbaised estimator of population mean Y¯ under stratified sampling scheme is as follows:

y¯st′=∑h=1LWhy¯h′.

The variance of y¯st′ is defined as

Vy¯st′=Y¯2δ2.0+∑h=1LWh2wS2yh2,
where w=N2hl−1Nhnh.

The customary ratio estimator under nonresponse is given by

y¯^Rst′=y¯st′pstP.

The MSE of y¯^Rst′ is given by

MSEy¯^Rst′=Y¯2δ2.0′+δ0.2−2δ1.1,
where δ2.0′=δ2.0+∑h=1LWh2wSyh22Y¯2.

The traditional Regression estimator under nonresponse is given by

y¯^stlr′=y¯st′+bstP−pst.

The MSE of y¯^stlr′ is

MSEy¯^stlr′=Y¯2δ2.0′1−ρst2,
where ρst=δ1.1δ2.0δ0.2.

Koyuncu [7] family of estimators for nonresponse is as follows:

y¯^kkst′=wc1y¯st′+wc2P−pstastP+bstastpst+bst.

Some family members of Koyuncu and Kadilar [7] in case of nonresponse are shown in Table 5.

y¯^kkst′ ast bst νj
y¯^kk1st′ 1 ψr ν3
y¯^kk2st′ 1 ψc ν2
y¯^kk3st′ ψr ψc ν4
y¯^kk4st′ ψc ψr ν5
Table 5

Some family members of Koyuncu and Kadilar [7] in case of nonresponse.

The minimum MSE of y¯^kkst′ is

MSEminy¯^kkst′=Y¯2−BkkDkk2+Akk′Ekk2−2CkkDkkEkkAkk′Bkk−Ckk2,
where
Akk′=Y¯21+δ2.0′+3νj2δ0.2−4νjδ1.1.

6.1. Adapted Estimators Under Nonresponse

y¯^BTst in case of nonresponse is as follows:

y¯^BTst′=y¯st′expP−pstP+pst.

The MSE of y¯^BTst′ for the first order approximation is given below:

MSEy¯^BTst′=Y¯2δ2.0′+14δ0.2−δ1.1.

y¯^di=y¯st in case of nonresponse is as follows:

y¯^′di=y¯′stcpstPm1wd+1−wdpstPm2m3.  

y¯^kd for nonresponse is as follows:

y¯^kd′=y¯st′pstPm4wkd+1−wkdastpst+bstastP+bstm5m6.  

In case of nonresponse, y¯^kd′, y¯^di′, and y¯^stlr′ also have equal MSE.

y¯^kist′ estimator for nonresponse situation is as follows:

y¯^kist′=ky¯staP+bαastpst+bst+1−αastP+bstM.  fori=1,2,…,6.

Some family members of Koyuncu and Kadilar [4] under nonresponse are shown in Table 6.

y¯^k′ νj
y¯^k1st′ ν1=PP+ψs
y¯^k2st′ ν2=PP+ψc
y¯^k3st′ ν3=PP+ψr
y¯^k4st′ ν4=ψrPψrP+ψc
y¯^k5st′ ν5=ψcPψcP+ψr
y¯^k6st′ ν6=ψrPψrP+ψs
Table 6

Some family members of Koyuncu and Kadilar [4] under nonresponse.

The minimum mean squared error of y¯^kist′ is

MSEminy¯^kist′=Y¯21−Ak′24Bk,
where
Ak′=M2+Mα2νj2δ2.0′−2Mανjδ1.1+2.

y¯^sgst in case of nonresponse is as follows:

y¯^sgst′=wsg1y¯st′+wsg2P−pstexpA¯−a¯stA¯+a¯st.

The minimum MSE of y¯^sgst′ is

MSEminy¯^sgst′=Y¯2−BsgDsg2+Asg′Esg24−CsgDsgEsgAsg′Bsg−Csg2,
where
Asg′=Y¯21+δ2.0+δ0.2(1+N)2−2δ1.11+N.

Family members of Koyuncu [6] under nonresponse are shown in Table 7.

y¯^nk′ Ust Vst γ′
y¯^nk1st′ ψb ψc 1
y¯^nk2st′ ψb ψc 2
y¯^nk3st′ ψc ψb 1
y¯^nk4st′ ψc ψb 2
y¯^nk5st′ ψr ψc 1
y¯^nk6st′ ψr ψc 2
Table 7

Family members of Koyuncu [6] under nonresponse.

y¯^nkst family of estimators under nonresponse as

y¯^nkst′=w1nky¯st′+w2nkpstPγ′expUstP−pstUstP+pst−2Vst.

The minimum MSE of y¯^nkst′ is

MSEminy¯^nkst′=Y¯2−BnkDnk2+Ank′Enk2−2CnkDnkEnkAnk′Bnk−Cnk2,
where
Ank′=Y¯21+δ2.0′+θnk2δ0.2−2θnkδ1.1.

6.2. Proposed Class of Estimators Under Nonresponse

Proposed class of estimators in case of nonresponse is as follows:

y¯^Nst′=ψ1″astP+bstβastpst+bst+1−βastP+bstg,
where
ψ1″=w1y¯st′2Ppst+pstP+w2pstPγ.

Some family members of proposed class under nonresponse are shown in Table 8.

y¯^Nst′ γ g ast bst β
y¯^N1st′ 12 1 ψb ψc 1
y¯^N2st′ 12 1 ψc ψb 1
y¯^N3st′ 12 1 1 ψr 1
y¯^N4st′ 12 1 1 ψb 1
y¯^N5st′ 12 1 1 ψc 1
y¯^N6st′ 13 1 ψc ψr 1
y¯^N7st′ 12 1 1 ψb 1
y¯^N8st′ 1 2 ψc ψr 1
y¯^N9st′ 1 2 ψr ψc 1
Table 8

Some family members of proposed class under nonresponse.

The minimum MSE of y¯^Nst′ is

MSEminy¯^Nst′=Y¯2L−AN′BN′,
where AN′=ϕBϕD2+ϕA′ϕE24−ϕCϕDϕE and BN′=ϕA′ϕB−ϕC2.

Further,

ϕA1′=Y¯21+δ2.0′+a2+2b+1δ0.2−2aδ11.

7. EFFICIENCY COMPARISON

In current section, we will compare the proposed estimators with the reviewed estimators exhibited in this study in light of the MSE of every estimator.

  • From Eqs. (25) and (3), MSEy¯^Rst−MSEminy¯^Nst>0 if

    ANBN−Y¯2L−δ2.0+δ0.2−2δ1.1>0.

  • From Eqs. (25) and (5), MSEy¯^stlr−MSEminy¯^Nst>0 if

    ANBN−Y¯2L−δ2.01−ρst2>0.

  • From Eqs. (25) and (8), MSEminy¯^kkst−MSEminy¯^Nst>0 if

    ANBN−BkkDkk2+AkkEkk2−2CkkDkkEkkAkkBkk−Ckk2−Y¯2L−1>0.

  • From Eqs. (25) and (10), MSEy¯^BTst−MSEminy¯^Nst>0 if

    ANBN−Y¯2L−δ2.0+14δ0.2−δ1.1>0.

  • From Eqs. (25) and (14), MSEminy¯^kist−MSEminy¯^Nst>0 if

    ANBN−Y¯2L−1+Ak2Bk>0.

  • From Eqs. (25) and (17), MSEminy¯^sgst−MSEminy¯^Nst>0 if

    ANBN−BsgDsg2+AsgEsg24−CsgDsgEsgAsgBsg−Csg2−Y¯2L−1>0.

  • From Eqs. (25) and (20), MSEminy¯^nkst−MSEminy¯^Nst>0 if

    ANBN−BnkDnk2+AnkEnk24−CnkDnkEnkAnkBnk−Cnk2−Y¯2L−1>0.

From the abovementioned conditions, we can argue that proposed estimators are more efficient than the existing ones in absence of nonresponse. Further, one can also develop such type of efficiency conditions for the nonresponse case.

8. NUMERICAL ILLUSTRATION

Here we assess the merits of proposed estimator over existing ones on the premise of percentage relative efficiency (PRE).

8.1. Population

The data are taken from Koyuncu [7]; where number of teachers consider as Y and for auxiliary attribute ϕ we use number of students classifying more (>) or less (<) than 750, in both primary and secondary schools as auxiliary variable for 923 districts at six regions in Turkey in 2007. For more detail see Koyuncu [7]. Some important descriptives are available in Table 9.

N1st=127 N2st=117 N3st=103 N4st=170 N5st=205
N6st=201 n1st=31 n2st=21 n3st=29 n4st=38
n5st=22 n6st=39 W1st=0.11375 W2st=0.1267 W3st=0.1115
W4st=0.1841 W5st=0.2221 W6st=0.2177 P1st=0.952 P2st=0.974
P3st=0.932 P4st=0.888 P5st=0.912 P6st=0.950 Y¯1st=703.74
Y¯2st=413 Y¯3st=573.17 Y¯4st=424.66 Y¯5st=267.03 Y¯6st=393.84
Cϕ1st=0.223 Cϕ2st=0.163 Cϕ3st=0.271 Cϕ4st=0.355 Cϕ5st=0.311
Cϕ6st=0.229 Cy1st=1.256 Cy2st=1.562 Cy3st=1.803 Cy4st=1.909
Cy5st=1.512 Cy6st=1.807 ρ1st=0.936 ρ2st=0.066 ρ3st=0.143
ρ4st=0.174 ρ5st=0.183 ρ6st=0.120 ρst=0.141 n=180
Table 9

Descriptives of population for absence of nonresponse.

Further, we consider 10%, 20%, and 30% values for nonresponse. Some important calculations related to nonresponse are available in Table 10. The PRE of the proposed and existing estimators for absence and presence of nonresponse w.r.t Vy¯st and Vy¯st′, respectively, available in Tables 11–14.

10%
N21st=13 S2y1st2=260681.7436 N22st=12 S2y2st2=149590.9697
N23st=11 S2y3st2=3210544.9636 N24st=17 S2y4st2=1736239.066
N25st=21 S2y5st2=76837.2619 N26st=20 S2y6st2=71170.42105
20%
N21st=25 S2y1st2=157424.9167 N22st=23 S2y2st2=164954.0198
N23st=21 S2y3st2=2737122.4000 N24st=34 S2y4st2=1773172.939
N25st=41 S2y5st2=117155.4524 N26st=40 S2y6st2=224079.7276
30%
N21st=38 S2y1st2=250264.1031 N22st=35 S2y2st2=127413.0672
N23st=31 S2y3st2=1914712.2796 N24st=51 S2y4st2=1451756.296
N25st=62 S2y5st2=90337.04733 N26st=60 S2y6st2=271760.609
Table 10

Some important results for nonresponse.

Estimator PRE Estimator PRE Estimator PRE
y¯st 100 y¯^kk1st 103.0878 y¯^nk6st 1356.459
y¯Rst 101.6047 y¯^kk2st 103.0877 y¯^N1st 19242.52
y¯BTst 101.5375 y¯^kk3st 103.0875 y¯^N2st 19020.75
y¯^stlr 101.7598 y¯^kk4st 103.0876 y¯^N3st 32694.96
y¯^k1st 103.0875 y¯^sgst 103.0874 y¯^N4st 23249.10
y¯^k2st 103.08750 y¯^nk1st 17514.15 y¯^N5st 57329.18
y¯^k3st 103.0667 y¯^nk2st 1961.927 y¯^N6st 48330.94
y¯^k4st 102.496 y¯^nk3st 4500.757 y¯^N7st 61076.41
y¯^k5st 103.0249 y¯^nk4st 1154.523 y¯^N8st 59545.86
y¯^k6st 102.4960 y¯^nk5st 6346.253 y¯^N9st 38961.67

PRE, percentage relative efficiency.

Table 11

PRE of proposed and existing estimators in absence of nonresponse.

Estimator PRE Estimator PRE Estimator PRE
y¯st′ 100 y¯^kk1st′ 103.0542 y¯^nk6st′ 1214.55
y¯′Rst 101.4037 y¯^kk2st′ 103.0541 y¯^N1st′ 17167.73
y¯′BTst 101.3449 y¯^kk3st′ 103.0539 y¯^N2st′ 16970.11
y′^stlr 101.53 y¯^kk4st′ 103.054 y¯^N3st′ 29169.73
y¯^k1st′ 103.054 y¯^sgst′ 103.0538 y¯^N4st′ 20742.58
y¯^k2st′ 103.053 y¯^nk1st′ 15686.5 y¯^N5st′ 51204.11
y¯^k3st′ 103.0365 y¯^nk2st′ 1752.667 y¯^N6st′ 43133.63
y¯^k4st′ 102.5359 y¯^nk3st′ 4015.272 y¯^N7st′ 54509.73
y¯^k5st′ 102.9984 y¯^nk4st′ 1034.06 y¯^N8st′ 53122.13
y¯^k6st′ 102.5359 y¯^nk5st′ 5661.139 y¯^N9st′ 34758.74
Table 12

PRE of proposed and existing estimators for 10% nonresponse.

Estimator PRE Estimator PRE Estimator PRE
y¯′st 100 y¯^kk1st′ 103.0692 y¯^nk6st′ 1356.459
y¯Rst′ 101.2502 y¯^kk2st′ 103.0691 y¯^N1st′ 19242.52
y¯BTst′ 101.198 y¯^kk3st′ 103.0688 y¯^N2st′ 19020.75
y¯^stlr′ 101.3705 y¯^kk4st′ 103.069 y¯^N3st′ 32694.96
y¯^k1st′ 103.069 y¯^sgst′ 103.0687 y¯^N4st′ 23249.10
y¯^k2st′ 103.0689 y¯^nk1st′ 17514.15 y¯^N5st′ 57392.18
y¯^k3st′ 103.054 y¯^nk2st′ 1961.927 y¯^N6st′ 48330.94
y¯^k4st′ 102.6068 y¯^nk3st′ 4500.757 y¯^N7st′ 61076.41
y¯^k5st′ 103.0187 y¯^nk4st′ 1154.523 y¯^N8st′ 59545.86
y¯^k6st′ 102.6068 y¯^nk5st′ 6346.253 y¯^N9st′ 38961.67

PRE, percentage relative efficiency.

Table 13

PRE of proposed and existing estimators for 20% nonresponse.

Estimator PRE Estimator PRE Estimator PRE
y¯′st 100 y¯^kk1st′ 103.1002 y¯^nk6st′ 1456.213
y¯′Rst 101.1609 y¯^kk2st′ 103.1001 y¯^N1st′ 20700.97
y¯′BTst 101.1125 y¯^kk3st′ 103.0998 y¯^N2st′ 20462.23
y¯^stlr′ 101.2726 y¯^kk4st′ 103.1 y¯^N3st′ 35172.98
y¯^k1st′ 103.1001 y¯^sgst′ 103.0997 y¯^N4st′ 25011.03
y¯^k2st′ 103.0999 y¯^nk1st′ 18799.27 y¯^N5st′ 61742.00
y¯^k3st′ 103.0865 y¯^nk2st′ 2109.021 y¯^N6st′ 51984.30
y¯^k4st′ 102.6703 y¯^nk3st′ 4842.024 y¯^N7st′ 65692.43
y¯^k5st′ 103.0528 y¯^nk4st′ 1239.21 y¯^N8st′ 64061.32
y¯^k6st′ 102.6703 y¯^nk5st′ 6827.846 y¯^N9st′ 41916.07

PRE, percentage relative efficiency.

Table 14

PRE of proposed and existing estimators for 30% nonresponse.

9. CONCLUSION

In this paper, we propose an estimator for the estimation of population mean Y under stratified random sampling scheme. The execution (performance) of the proposed estimator is assessed theoretically and numerically using natural population. We also consider 10%, 20%, and 30% values for nonrespondents. The results of PRE showed that the proposed estimator is more efficient as compare to the customary ratio, regression, Bahl and Tuteja [1], Diana [2], Kadilar and Cingi [14], Koyuncu and Kadilar [3], Koyuncu and Kadilar [4], Koyuncu and Kadilar [10], Shabbir and Gupta [5], and Koyuncu [6] for both situations. Hence, it is advisable to utilize the proposed class of estimators for the estimation of population mean under stratified random sampling scheme when the information of auxiliary attribute is known.

REFERENCES

3.N. Koyuncu and C. Kadilar, Pak. J. Statist., Vol. 26, No. 2, 2010, pp. 427-443.
5.J. Shabbir and S. Gupta, Commun. Stat. Theory Methods, Vol. 40, No. 2, 2011, pp. 199-212.
7.N. Koyuncu, Proceeding of 59-th ISI world Statistics Congress, pp. 25-30.
11.B.V.S. Sisodia and V.K. Dwivedi, J. Indian Soc. Agric. Stat., Vol. 33, 1981, pp. 13-18.
12.H.P. Singh and M.S. Kakran, Unpublished Paper, 1993.
17.M.K. Chaudhary, R. Singh, R.K. Shukla, M. Kumar, and F. Smarandache, Pak. J. Stat. Oper. Res., Vol. 5, No. 1, 2009, pp. 47-54.
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
18 - 1
Pages
12 - 25
Publication Date
2019/04/22
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.190306.002How to use a DOI?
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risenwbib
TY  - JOUR
AU  - Usman Shahzad
AU  - Muhammad Hanif
AU  - Nursel Koyuncu
AU  - A. V. Garcia Luengo
PY  - 2019
DA  - 2019/04/22
TI  - A Family of Ratio Estimators in Stratified Random Sampling Utilizing Auxiliary Attribute Along Side the Nonresponse Issue
JO  - Journal of Statistical Theory and Applications
SP  - 12
EP  - 25
VL  - 18
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190306.002
DO  - 10.2991/jsta.d.190306.002
ID  - Shahzad2019
ER  -