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Volume 17, Issue 4, December 2018, Pages 616 - 626

A Generalized Class of Estimators for Estimating Population Mean in the Presence of Non-Response

Authors
Iram Saleem1, Aamir Sanaullah2, Muhammad Hanif1, *
1Department of Statistics, National College of Business Administration and Economics, Lahore, Pakistan
2Department of Statistics, COMSATS Institute of Information and Technology, Lahore, Pakistan.
*

Corresponding author. Email: dmianhanif@gmail.com

Received 28 June 2017, Accepted 13 January 2018, Available Online 31 December 2018.
DOI
10.2991/jsta.2018.17.4.4How to use a DOI?
Keywords
Auxiliary variable; Mean square error; Non-response; Stratified random sampling
Abstract

Koyuncu and Kadilar proposed an estimator based on single auxiliary variable with complete response in stratified random sampling. In this paper, we extended Koyuncu and Kadilar's estimator to a more generalized class of estimators using two-auxiliary variables in stratified random sampling for the situation of non-response and further introduced its another improved generalized class of estimators. The mathematical conditions under which proposed class of estimators are efficient as compare to Hansen and Hurwtiz estimator, and ratio estimators modified for stratified sampling have been derived. An empirical study has also been carried out to examine the performance of the suggested estimators.

Copyright
© 2018 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. INTRODUCTION

Non-response refers to the situation, when an investigator fails to get necessary information from some of the units of the selected sample. The problem of non-response was first analyzed by Hansen and Hurwitz [1]. They developed a classical non-response concept to obtain information from the sub-sample of non-response group. An estimator for the population mean in the presence of non-response was constructed and also derived its variance with the optimum sampling fraction for the non-respondents. It is suitable for the surveys, in which first attempt is made on mail questionnaires and second attempt is selected from the non-respondent persons by personnel interviews. Following Hansen–Hurwitz methodology Cochran [2] proposed ratio type estimator for dealing with non-response. Chaudhary et al. [3], Haq and Shabbir [4] and Sanaullah et al. [5] presented some improved estimators for stratified random sampling under non-response.

In order to improve the efficiency of an estimator, auxiliary information is often used to estimate the unknown population mean of study variable. Cochran [6] discussed classical ratio estimators. Further, Cochran [2], Kadilar and Cingi [7], Shabbir and Gupta [8], Koyuncu and Kadilar [9], Sanaullah et al. [10] and Sanaullah et al. [11] utilized auxiliary information under stratified random sampling scheme.

2. SAMPLING DESIGN AND PROCEDURE DEALING WITH NON-RESPONSE

Consider a finite population of size N is stratified into L homogenous strata. Let Nh be the size of hth stratum (h = 1,2,3,…, L) such that ∑h=1LNh=N and (yhi, xhi, zhi) be the observations of study variable y and auxiliary variables x and z on the ith unit of hth stratum, respectively. Let y¯h,x¯h and z¯h be the sample means of hth stratum corresponding to the population means Y¯h,X¯h and Z¯h respectively. Usually it is not possible to collect complete information from all the units selected in the sample nh∑h=1Lnh=n. Let nh(1) units from a sample of nh provide their responses and nh(2) units do not. Adapting Hansen and Hurwitz [2] sub-sampling methodology, a sub-sample of size rh(rh=nh(2)fh;fh>1) from nh(2) non-respondents group is selected at random and 1/fh denotes the sampling fraction among the non-respondent group in the hth stratum. In practice, rh is usually not integer and has to be rounded. In accordance with most of the current literature on the topic, let us assume that the followed-up rh(⊂nh(2)) units respond on the second call. Moreover, let Uh denote a dummy variable which takes value uhi on the ith population unit of stratum h and has mean U¯h. Hereafter, U¯h may stand for Yh; Xh or for a second auxiliary variable Zh. Let;

u¯nh(1)=∑i=1nh(1)ui(1)nh(1), u¯rh=∑i=1rh(1)ui(2)rh
and
u¯h*=nh(1)nhu¯(1)nh1+nh(2)nhu¯(2)rh
where u¯(1)nh(1) is mean of nh(1) respondents on first call and u¯(2)rh is mean of rh units respond on the second call, and u¯h* denotes the unbiased Hansen–Hurwitz [1] of U¯h for stratum h.

A modified Hansen and Hurwitz [1] unbiased estimator for stratified sampling may be given as,

t1=∑h=1LPhu¯h*,
The variance of t1 is,
Vart1=∑h=1LλhPh2Suh2+∑h=1Lλh*Ph2Suh(2)2.
where Suh2=∑i=1Nhuhi−U¯h2/Nh−1, and Suh(2)2=∑i=1Nh(2)ui−U¯h(2)2/Nh(2)−1 are the mean square error (MSE) of the entire group and the non-response group of the study variable with Ph=Nh/N,λh=1nh−1Nh,λh*=fh−1nhWh(2),Wh(2)=Nh(2)/Nh, and fh=nh(2)/rh.

The modified form of ratio and product estimators from stratified random sampling under non-response defined by Cochran [2] may be written as:

t2=y¯st*X¯x¯st*Z¯z¯st*,
and
t3=y¯st*x¯st*X¯z¯st*Z¯,
where y¯st* and x¯st* are Hansen–Hurwitz [1] estimators modified to the stratified sampling for population means X¯ and Y¯ respectively.

The MSE of the estimators t2 and t3 are given respectively as

MSEt2≈Y¯2∑h=1LλhShy2Y¯2+Shx2X¯2+Shz2Z¯2−2ShxyY¯X¯−2ShyzY¯Z¯+2ShyzX¯Z¯+λh*Shy(2)2Y¯2+Shx(2)2X¯2+Shz(2)2Z¯2−2Shxy(2)Y¯X¯−2Shyz(2)Y¯Z¯+2Shxz(2)X¯Z¯,
and
MSEt3≈Y¯2∑h=1LλhShy2Y¯2+Shx2X¯2+Shz2Z¯2+2ShxyY¯X¯+2ShyzY¯Z¯+2ShyzX¯Z¯+λh*Shy(2)2Y¯2+Shx(2)2X¯2+Shz(2)2Z¯2+2Shxy(2)Y¯X¯+2Shyz(2)Y¯Z¯+2Shxz(2)X¯Z¯.

Chaudhary et al. [3] presented ratio estimator for stratified sampling when non-response is present only on study variable as,

t4=y¯st*aX¯+bαax¯st+b+1−αaX¯+bg,
The MSE of t3 is
MSEt4≈∑h=1LPh2λhSyh2+α2v2g2R2Sxh2−2αvgRρxyhSxhSyh+∑h=1LPh2λh*Syh(2)2,
where
v=aX¯aX¯+b,R=Y¯X¯,

The MSE(t5) is minimum for

α=∑h=1LPh2λhρxyhSyhSxhvgR∑h=1LPh2λhSxh.

The aim of this paper is to propose a more generalized class of estimators for estimating population mean considering the non-response in stratified random sampling using two auxiliary variables. Also the purpose is to introduce another improved form of proposed generalised estimator. Another purpose is to determine the optimum size of the sample and the sub-sampling fractions of the non-respondent group for the fixed cost.

3. PROPOSED GENERALIZED CLASS OF ESTIMATORS

In this section, following Koyuncu and Kadilar [12] we proposed a generalized class of estimators for estimating a finite population mean in stratified random sampling considering the presence of non-response using two auxiliary variables as,

ta=y¯st*aXX¯+bXαXaXx¯st*+bX+1−αXaXX¯+bXgXaZZ¯+bZαZaZz¯st*+bX+1−αZaZZ¯+bZgZ,
where the constants ax & az(≠0), and bx & bz are either real numbers or the functions of the auxiliary variable, in form of coefficient of variations, standard deviations, correlation coefficients, skewness or kurtosis from the population, whereas gx and gZ are known constants take the value (0, 1, −1) to produce respectively unbiased estimator, different families of ratio-cum-ratio and families of product-cum-product type estimators, and αx and αz are the constants to be determined such that MSE of the proposed estimator ta is minimum. Taking different values of the constants we may obtain many families of ratio-cum-ratio and families of product-cum-product estimators such as some examples are presented in Table 1.

Family of ratio-cum-ratio gx=gz=1 Family of product-cum-product gx=gz=−1 aX bX aZ bZ αX αZ
ta1 ta2 1 0 1 0 1 1
ta3 ta4 1 ρxy 1 ρyz 1 1
ta5 ta6 σx 1 σz 1 1 1
ta7 ta8 ρxy 1 ρyz 1 1 1
Table 1

Families of the estimators ta.

3.1 Bias and MSE of Proposed Generalized Class of Estimator

To obtain the bias and MSE, we consider

y¯st*=∑h=1LPhy¯h*=Y¯1+e0*, x¯st*=X¯1+e1*, z¯st*=Z¯1+e2*
E(ei*)=0 for i=0,1,2
and
Vr,s,t,*=∑h=1LPhr+s+tEx¯h′*−X¯hry¯h*−Y¯hsz¯h*−Z¯htX¯rY¯sZ¯tEe0*2=1Y¯2∑h=1LPh2λhSyh2+λh*Syh22=V020*     Ee1*2=1X¯2∑h=1LPh2λhSxh2+λh*Sxh22=V200*Ee2*2=1Z¯2∑h=1LPh2λhSzh2+λh*Szh22=V002*     Ee0*e2*=1YZ¯∑h=1LPh2λhSyzh+λh*Syzh2=V011*Ee0*e1*=1YX¯∑h=1LPh2λhSxyh+λh*Sxyh2=V110*     Ee1*e2*=1ZX∑h=1LPh2λhSxzh+λh*Sxzh2=V101*

On rewriting we may get Eq. (11) as

ta=Y¯1+e0*1+αxνxe1*−gx1+αzνze2*−gz,
where
vx=axX¯axX¯+bx and vz=azZ¯azZ¯+bz.

We assume that |αxνxe1*|<1 and |αzνze2*|<1, so that we may expand the series, 1+αxνxe1*−gX and 1+αzνze2*−gz, we get

ta=Y¯1+e0*1−gxαxνxe1*+gxgx+12αxνx2e1*2+…    1−gzαzνze2*+gzgz+12αzνz2e2*2+….
It is assumed that the contribution of terms involving powers in e0*, e1* and e2* higher than two is negligible. We have,
ta−Y¯=Y¯e0*−gxαxνxe1*−gxαxνxe0*e1*+gxgx+12αxνxe1*2−gzαzνze12*−gzαzνze0*e2*+gxαxνxgzαzνze1*e2*+gzgz+12αzνz2e2*2.
Taking expectation of Eq. (15), we will get the bias as,
Biasta=gxαxνxV110*+gxgx+12αx2νx2V200*−gzαzνz V011*+gxαxνxgzαzνz V101*+gzgz+12αz2νz2 V002*2.
we take square of Eq. (15) and retain terms up to the order n−1 then we take expectation to get the MSE of the estimator ta as,
MSEta=Y¯2V020*+gxαxνx2V200*+gzαzνz2V002*−2gxαxνxV110*      −2gzαzνzV011*+2gxαxνxgzαzνzV101*.
Minimization of Eq. (17) with respect to αx and αz yields the optimum values as
αx=V011*V101*−V110*V002gxνxV101*2−V002*V200*, and αz=V101*V110*−V011*V200gzνzV101*2−V002*V200*.

Using the Eq. (18), the expression of minimum MSE may be obtained as

MSEta=Y¯2V020*−V002*V110*2+V200*V011*2−2V101*V011*V110*V200*V002*−V101*2.

For generalized family of ratio-cum-ratio estimators presented in Table 1, we can give the MSE expression in Eq. (17) as,

MSEtai=Y¯2[V020*+V200*+V002*−2V110*−2V011*+2V101*]=MSEt2 i=1Y¯2V020*+vx(i−12)2V200*+vz(i−12)2V002*−2vx(i−12)V110*−2vz(i−12)V011*+2vx(i−12)vz(i−12)V101* i=3,5,7
and for generalized family of product-cum-product estimators, the MSE expression can be given as,
MSEtaj=Y¯2[V020*+V200*+V002*+2V110*+2V011*+2V101*]=MSEt3 j=2Y¯2V020*+vx(j2−1)2V200*+vz(j2−1)2V002*+2vx(j2−1)V110*+2vz(j2−1)V011*+2vx(j2−1)vz(j2−1)V101* j=4,6,8

Where vx1=X¯X¯+ρxy and vz1=Z¯Z¯+ρyz

vx2=σxX¯σxX¯+1 and vz2=σzZ¯σzZ¯+1

vx3=ρxyX¯ρxyX¯+1 and vz3=ρyzZ¯ρyzZ¯+1

One can think many more estimators from Eq. (11), and the bias and MSE expression for these estimators can be expressed by Eqs. (16) and (17) respectively.

4. ANOTHER PROPOSED GENERALIZED ESTIMATORS:

In this section, we have shown another improved and generalized form for the estimator ta proposed in Section 3. The proposed estimator ts for estimating population mean is given as,

ts=ηy¯st*aXX¯+bXαXaXx¯st*+bX+1−αXaXX¯+bXgXaZZ¯+bZαZaZz¯st*+bX+1−αZaZZ¯+bZgZ
or may also be consider as,
ts=ηY¯1+e0*1+αxνxe1*−gx1+αzνze2*−gz,

We assume that |αxνxe1*|<1 and |αzνze2*|<1, so that we may expand the series, 1+αxνxe1*−gX and 1+αzνze2*−gz, we get

ts=ηY¯1+e0*1−gxαxνxe1*+gxgx+12αxνx2e1*2+…       1−gzαzνze2*+gzgz+12αzνz2e2*2+….

It is assumed that the contribution of terms involving powers in e0*, e1* and e2* higher than two is negligible. We have,

ts−Y¯=Y¯η−1+Y¯e0*−gxαxνxe1*−gxαxνxe0*e1*+gxgx+12αxνxe1*2−gzαzνze2*  −gzαzνze0*e2*+gxαxνxgzαzνze1*e2*+gzgz+12αzνz2e2*2.

The bias and MSE expressions of ts may be given respectively as,

Biasts=Y¯η−1+Biasta
and
MSEts=Y¯2η−12+η2MSEta+2Y¯ηη−1Biasta

We differentiate Eq. (25) w.r.t η˜, and equate it to zero, we get

η˜=Y¯Y¯+BiastaY¯2+MSEta+2Y¯Biasta,

On substitution the optimum value of η as given in Eq. (26), in the result Eq. (25), the minMSE of the proposed estimator ts is obtained as,

min MSEts=Y¯2−Y¯2Y¯+Biasta2Y¯2+MSEta+2Y¯Biasta
Or
min MSEts=Y¯21−1+BiastaY¯21+MSEtaY¯2+2BiastaY¯

We can think many more improved estimators taking different values of the constants in Eq. (20), such as some example are given in Table 2.

Family of ratio-cum-ratio gx=gz=1 Family of product-cum-product gx=gz=−1 (ax,bx,az,bz) αx αz η
ts1 ts2 (1,0,1,0) 1 1 η
ts3 ts4 (1,ρxy,1,ρyz) 1 1 η
ts5 ts6 (σx,1,1,σy) 1 1 η
ts7 ts8 (ρxy,1,1,ρyz) 1 1 η
Table 2

Families of the proposed estimator ts.

The MSE of the ratio-cum-ratio and product-cum-product estimators given in Table 2 can be given using Eq. (27) as,

MSEtsi=Y¯21−1+BiastaiY¯21+MSEtaiY¯2+2BiastaiY¯ for i=1,3,5,7
MSEtsj=Y¯21−1+BiastajY¯21+MSEtajY¯2+2BiastajY¯ for j=2,4,6,8

5. MATHEMATICAL COMPARISON

In this section, the MSE of the suggested family of estimator has been compared with the mean estimator, stratified ratio estimator and the class of estimators. Let us consider following notations as:

Ai=vx(i−12)2V200*+vz(i−12)2V002*−2vx(i−12)V110*−2vz(i−12)V011*+2vx(i−12)vz(i−12)V101* i=3,5,7
Aj=vx(j2−1)2V200*+vz(j2−1)2V002*+2vx(j2−1)V110*+2vz(j2−1)V011*+2vx(j2−1)vz(j2−1)V101* j=4,6,8
B1=vz(i−12)2V002*−2vz(i−12)V011*−V200*−V002*+2V110*+2V011*−2V101*
B2=V002*V110*2+V200*V011*2−2V101*V011*V110*V200*V002*−V101*2
and
Ci=1+BiastaiY¯21+MSEtaiY¯2+2BiastaiY¯

The efficiency conditions may be written as:

  1. Vart1>MSEtai  i=3,5,7IfV020*−Ai>0

  2. MSEt2>MSEtaiIfmin−Ai±Ai2−B1V200*V200*≤vx(i−12)≤max−Ai±Ai2−B1V200*V200*

  3. Vart1>min MSEtaiIfB2>0

  4. MSEt2>min MSEtaiIfB2>V200*+V002*−2V110*−2V011*+2V101*

  5. Vart1>MSEtsiIFCi>1−V020*

  6. MSEt2>MSEtsiIfCi>1−V020*+V200*+V002*−2V110*−2V011*+2V101*

  7. min MSEtai>min MSEtsiIfCi>1+B2−V020*

6. COST FUNCTION AND SAMPLE SIZE ESTIMATION

Let us assume a linear cost function, the total cost of the sample survey could be written as

C′=∑h=1Lch0nh+∑h=1Lch1nh+∑h=1Lch2rh
where ch0 denotes the per unit cost of making first attempt, ch1 denotes per unit cost for processing the result of all characteristics in first attempt and ch2 denotes the per unit cost for processing the result of all characteristics in second attempt in the hth stratum.

The total expected cost of the survey could be given as

C=EC′=∑h=1Lch0+ch1Wh1nh+∑h=1Lch2rh

6.1. Cost Function for Estimator ta

Let us consider the function for estimator ta

φnh,rh,δ=Varta+δC−C0

φnh,rh,δ=Y¯2V020*+uX2V200*+uZ2V002*−2uXV110*−2uZV011*+2uXuZV101*+δ∑h=1Lch0+ch1Wh1nh+∑h=1Lch2rh−C0

Let

Sh=1Y¯2Syh2+1X¯2Sxh2+1Z¯2Szh2−2ux1X¯Y¯Sxyh−2uz1Y¯Z¯Syzh+2uxuz1X¯Z¯Sxzh
and
Sh*=1Y¯2Syh*2+1X¯2Sxh*2+1Z¯2Szh*2−2ux1X¯Y¯Sxyh*−2uz1Y¯Z¯Syzh*+2uxuz1X¯Z¯Sxzh*

So we can write Eq. (39) as

φnh,rh,δ=Y¯2∑h=1LPh21nh−1NhSh+W2hfh−1rhSh*+δ∑h=1Lch0+ch1Wh1nh+∑h=1Lch2rh−C0

Where δ is lagrangian multiplier. On differentiating Eq. (41) we can get the values of nh, rh, and δ as,

nh=Y¯2Ph2Shδch0+ch1Wh1

Also

rh=Y¯2Ph2W2hfh−1Sh*δch2
C0Y¯∑h=1LPhch0+ch1Wh1Sh+W2hfh−1Sh*ch2=1δ

Let

D=∑h=1LPhch0+ch1Wh1Sh+W2hfh−1Sh*ch2
1δ=C0Y¯D

Subsituting δ from Eqs. (44) to (42), we have

nh′=C0Ph2ShDch0+ch1Wh1

Similarly, substitute the value of δ from Eqs. (44) to (43) we have

rh′=C0Ph2W2hfh−1Sh*Dch2

7. EMPIRICAL STUDY

In order to see the performance of the suggested family of estimators as compare to class of estimators under stratified random sampling. The statistics of the two stratified populations have been given in Table 3.

Population-I: [Source: Koyuncu and Kadilar [9]]

We consider No. of teachers as study variable (Y), No. of students as auxiliary variable (X), and No. 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 (6 940) male and (1 678) female households in flood affected districts of Pakistan.

Neyman allocation has been used in order to allocate sample sizes to different strata in the two populations separately as:

nh=nNhSh∑h=1LNhSh

From Table 3, we observe a positive correlation among study variable and the auxiliary variable in order to use ratio estimators for estimating population mean. The comparison of the proposed estimators have been made with respect to Hansen-Hurwtiz [1] and ratio estimators modified for stratified sampling. The information for the two stratified populations is given in Table 3.

Stratum (h) Population-I Population-II
1 2 3 4 5 6 1 2 3
Stratified mean, S.Ds and correlation coefficients Nh 127 117 103 170 205 201 21 34 26
nh 31 21 29 38 22 39 06 04 02
n′h 70 50 75 95 70 90 15 17 08
Syh 883.84 644.92 1 033.40 810.58 403.65 711.72 12.14 8.34 5.47
Sxh 30 486.7 15 180.77 27 549.69 18 218.93 8 497.77 23 094.14 76.71 31.94 49.55
Szh 555.58 365.46 612.95 458.03 260.85 397.05 19.48 07.10 13.21
Y¯h 703.74 413 573.17 424.66 267.03 393.84 37.55 37.25 26.39
X¯h 20 804.59 9 211.79 14 309.30 9 478.85 5 569.95 12 997.59 116.57 093.00 26.39
Z¯h 498.28 318.33 431.36 311.32 227.20 313.71 114.14 106.50 118.88
ρxyh 0.9360 0.9960 0.9940 0.9830 0.9890 0.9650 0.7914 0.8339 0.7696
ρxzh 0.9396 0.9696 0.9770 0.9640 0.9670 0.9960 0.9894 0.8820 0.9669
ρyzh 0.9790 0.9760 0.9840 0.9830 0.9640 0.9830 0.7781 0.6651 0.5935
Wh = 10%
Non-response		 Syh2 510.57 386.77 1 872.88 1 603.30 264.19 497.84 08.66 10.05 03.95
Sxh2 9 446.93 9 198.29 52 429.99 34 794.9 4 972.56 12 485.10 42.14 13.28 74.22
Szh2 303.92 278.51 960.71 821.29 190.85 287.99 6.25 5.20 20.53
ρxy2 0.9961 0.9975 0.9998 0.9741 0.9950 0.9284 0.9997 0.9995 0.9840
ρxz2 0.9901 0.9895 0.9964 0.9609 0.9865 0.9752 0.9707 1.0000 0.9999
ρyz2 0.9931 0.9871 0.9972 0.9942 0.9850 0.9647 0.9649 0.9996 0.9819
Wh = 20%
Non-response		 Syh2 396.77 406.15 1 654.40 1 333.35 335.83 903.91 7.96 8.47 4.06
Sxh2 7 439.16 8 880.46 4 5784.78 2 9219.30 6 540.43 28 411.44 36.50 25.82 59.32
Szh2 244.56 274.42 965.42 680.28 214.49 469.86 5.20 8.18 16.54
ρxy2 0.9954 0.9931 0.9960 0.9761 0.9966 0.9869 0.9905 0.8026 0.8601
ρxz2 0.9897 0.9884 0.9789 0.9629 0.9820 0.9825 0.9623 0.9858 0.9956
ρyz2 0.9898 0.9798 0.9846 0.9940 0.9818 0.9874 0.9297 0.8062 0.8112
Wh = 30%
Non-response		 Syh2 500.26 356.95 1 383.70 1 193.47 289.41 825.24 12.70 09.86 4.50
Sxh2 14 017.99 7 812.00 38 379.77 26 090.60 5 611.32 24 571.95 37.69 24.02 52.26
Szh2 284.44 247.63 811.21 631.28 188.30 437.90 9.42 6.83 14.54
ρxy2 0.9639 0.9919 0.9955 0.9801 0.9961 0.9746 0.9288 0.8335 0.8275
ρxz2 0.9107 0.9848 0.9771 0.9650 0.9794 0.9642 0.9062 0.8859 0.9907
ρyz2 0.9739 0.9793 0.9839 0.9904 0.9799 0.9829 0.9696 0.5877 0.7542
Table 3

Data statistics for the two populations.

The MSE values of the proposed class of estimators are computed in Table 4 two different data sets. The percent relative efficiencies of the estimators are given in Table 5. Efficiency of each estimator has been tested by increasing the non-response rate from 10% to 30% each with three different values of fh (2, 2.5 & 3). From Table 4 it is observed that MSE's of the estimators are increased if non-response increases from 10% to 30% however from Table 5 it is noticed that PRE of the proposed estimator is also increased if non-response increases from 10% to 30% which shows that the proposed estimators ta, and ts as compare to the existing estimators (t1 and t2) can perform more efficiently even at higher non-response rate. If we compare proposed estimators ta and ts with each other, we observe that MSE values of the proposed estimators ts are smaller than the MSE values of proposed estimator ta at each non-response rate. If we compare the two proposed estimators ta and ts on basis their PRE values, the conclusion can be drawn that ts is an improved form of ta.

MSEs of the different existing estimators and Proposed class of estimators
Wh fh t1 t2 tamin tsmin
2 2 143.99 1 411.42 76.1190 76.0883
10% 2.5 2 370.94 1 501.89 81.7661 81.7304
3 2 597.83 1 592.34 87.1097 87.0691
2 2 540.33 1 667.89 83.2651 83.2284
20% 2.5 2 965.45 1 886.62 92.3106 92.2641
3 3 390.54 2 105.42 101.171 101.116
2 2 703.09 1 753.48 88.1595 88.1179
30% 2.5 3 209.59 2 015.08 99.6831 99.6291
3 3 716.08 2 276.639 110.882 110.815
Table 4

MSEs values of estimators.

PREs of the different existing estimators and Proposed class of estimators
Wh fh t1 t2 tamin tsmin
2 100 151.90305 2 816.6292 2 817.7657
10% 2.5 100 157.86376 2 899.6614 2 900.9279
3 100 163.14543 2 982.2511 2 983.6417
2 100 152.30801 3 050.8941 3 052.2394
20% 2.5 100 157.18322 3 212.4696 3 214.0887
3 100 161.03865 3 351.2963 3 353.1192
2 100 154.15574 3 066.1358 3 067.5833
30% 2.5 100 159.27854 3 219.7935 3 221.5387
3 100 163.22658 3 351.3826 3 353.4088
Table 5

PRE values of estimators.

8. CONCLUSION

From the results of the numerical study, we infer that both the proposed estimators using auxiliary information perform more efficiently (with substantial gain in precision) than t1 and t2. Hence proposed estimators are recommended for their practical use.

REFERENCES

1.M.H. Hansen and W.N. Hurwitz, J. Am. Stat. Assoc., Vol. 41, No. 236, 1946, pp. 517-529.
2.W.G. Cochran, Sampling Techniques, Third ed., John Wiley, New York, 1977.
3.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.
4.A. Haq and J. Shabbir, Commun. Stat. Theory Methodol., Vol. 42, No. 1, 2013, pp. 782-799.
5.A. Sanaullah, M. Noor-ul-Amin, and M. Hanif, Pak. J. Stat., Vol. 31, No. 1, 2015, pp. 71-94.
6.W.G. Cochran, J. Am. Stat. Assoc., Vol. 37, 1942, pp. 199-212.
7.C. Kadilar and H. Cingi, Biom. J., Vol. 45, No. 2, 2003, pp. 218-225.
8.J. Shabbir and S. Gupta, Am. J. Math. Manag. Sci., Vol. 25, No. 3-4, 2005, pp. 293-311.
9.N. Koyuncu and C. Kadilar, J. Stat. Plan. Inference, Vol. 139, No. 8, 2009b, pp. 2552-2558.
10.A. Sanaullah, H. Khan, H.A. Ali, and R. Singh, J. Reliab. Stat. Stud., Vol. 5, No. 2, 2012, pp. 119-132.
11.A. Sanaullah, H.A. Ali, M. Noor ul Amin, and M. Hanif, Appl. Math. Comput., Vol. 226, 2014, pp. 541-547.
12.N. Koyuncu and C. Kadilar, Commun Stat Theory Methodol., Vol. 38, No. 1, 2009a, pp. 2398-2417.
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 4
Pages
616 - 626
Publication Date
2018/12/31
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.2018.17.4.4How to use a DOI?
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risenwbib
TY  - JOUR
AU  - Iram Saleem
AU  - Aamir Sanaullah
AU  - Muhammad Hanif
PY  - 2018
DA  - 2018/12/31
TI  - A Generalized Class of Estimators for Estimating Population Mean in the Presence of Non-Response
JO  - Journal of Statistical Theory and Applications
SP  - 616
EP  - 626
VL  - 17
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.4.4
DO  - 10.2991/jsta.2018.17.4.4
ID  - Saleem2018
ER  -