Generalized Semi Exponential Type Estimator under Systematic Sampling

Muhammad Nouman Qureshi; Sadia Khalil; Muhammad Hanif

doi:10.2991/jsta.2018.17.2.8

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Volume 17, Issue 2, June 2018, Pages 283 - 290

Generalized Semi Exponential Type Estimator under Systematic Sampling

Authors

Muhammad Nouman Qureshi¹^{, 3}^,nqureshi633@gmail.com, Sadia Khalil²^,sadia_khalil@hotmail.com, Muhammad Hanif³^,drmianhanif@gmail.com

¹School of Statistics, University of Minnesota, Minneapolis-55455, USA

²Department of Statistics, Lahore College for Women University, Lahore-54000, PAKISTAN

³Department of Statistics, National College of Business Administration & Economics, Lahore-54660, PAKISTAN

Available Online 30 June 2018.

DOI: 10.2991/jsta.2018.17.2.8 How to use a DOI?
Keywords: Auxiliary information; Generalized Semi-Exponential type estimator; Systematic Sampling; Mean Square Error; Percentage Relative Efficiency
Abstract: In sample surveys, collection of auxiliary information together with the main variable of interest is very important to increase the efficiency of the estimators of population parameters of interest. Regression and ratio estimation are very popular and are widely used methods that benefit from the use of auxiliary information for the estimation of population parameters like mean, total, variance, proportion etc. A generalized semi-exponential type estimator is proposed in this paper using two auxiliary variables under the framework of systematic sampling. The expressions of approximate bias and mean square error of the proposed estimator have been derived. Algebraic conditions have been obtained under which the proposed estimator is more efficient than the competing estimators considered here. An empirical study has been carried out to show the improvement in efficiency of the proposed estimator as compared to the existing estimators.
Copyright: Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

In survey sampling, no single estimation procedure will always work the best. Various sampling designs are available for different populations under different situations. The natural variation among real life situations requires development of different types of estimators for different sampling techniques under different situations for different populations.

Systematic sampling is often advantageous over the simple random sampling for being easy to implement and often providing increased precision in estimates of population parameters of interest. Under this design, only first sampling unit is selected randomly and the subsequent units are then selected by according to certain rules. According to W.G. Madow and L.H. Madow (1944),”Systematic sampling is most commonly used probability design for the estimation of finite population parameters, due to its simplicity. ”Cochran (1946) declared that apart from easy to implement, systematic sampling often provides more efficient estimators as compared to the simple random sampling (SRS) or stratified random sampling for different types of populations under different situations.

Auxiliary information is commonly used together with the main variable of interest to improve the estimates of population parameter like the mean, total and variance, etc. Ratio and regression estimation methods use auxiliary information in many ways to obtain better estimation results in terms of minimum mean square errors. Various authors have used auxiliary information to improve the estimators in terms of relative efficiency under systematic sampling. For details, readers may refer to Quenouille (1956) and Hansen et al. (1946). Swain (1964), Shukla (1971) and Singh (1967) have proposed the classical ratio, product and ratio-cum-product-type estimators respectively under the framework of systematic sampling. Srivastava and Jhajj (1983) used multi-auxiliary variables to propose a new family of estimators. Kushwaha and Singh (1989), Banarasi et al. (1993) and Singh and Singh (1998) suggested different modified ratio, product and difference type estimators under systematic sampling. For more recent work on systematic sampling including some exponential type estimators using auxiliary information, one can refer to Singh et al. (2011), Singh and Solanki (2012), Singh and Jatwa (2012),Tailor et al.(2013), Khan and Singh (2015) and Khan (2016).

In this paper, a generalized semi-exponential type estimator is proposed with two auxiliary variables with the expectation that this estimator will be more efficient than some of the other competing estimators considered in this paper under systematic sampling. Some existing estimators along with the methodology and useful notations of systematic sampling are given in Section 2. The expressions for approximate bias and mean square error of the proposed estimator have been derived in Section 3. Some special cases are also given in the same Section in which the proposed estimator reduces to many other exponential and non-exponential type estimators. Theoretical comparisons with some other existing estimators are addressed in Section 3. An empirical study that shows percentage relative efficiency of the proposed estimator with respect to the mean per unit estimator under systematic sampling is carried out in Section 4. Some concluding remarks are given in Section 5.

2. Methodology of Systematic Sampling with Associated Estimators

In this section, we introduce the following terminology that is needed in this paper. Let y is the study variable and x and z be the auxiliary variables defined on a finite population P consisting of N distinct but identifiable units, P = (P₁, P₂, … , P_N), numbered in some specific order. A random sample of size n is selected from the first k units and then every k^th units is selected corresponding to each unit in the sample from the first k units. So there will be total k samples, each of size n, such that N = nk, where n and k are positive integers. Let (y_ij, x_ij, z_ij) for i=1,2,3,…, k and j=1,2,3,…,n denote the values of j^th unit in the i^th sample. The systematic sample means of the variable of interest and the auxiliary variables to be estimated as, y¯sys=n−1∑j=1nyij , x¯sys=n−1∑j=1nxij and z¯sys=n−1∑j=1nzij are unbiased estimators of the corresponding population means Y¯ , X¯ and Z¯ respectively.

We also denote the following error terms and the other notations:

(1)y¯sys=Y¯(1+ey) and z¯sys=Z¯(1+ez)such that E(ey)=E(ex)=E(ez)=0E(ey2)=θ(1+(n−1)ρy)Cy2=C02,E(ex2)=θ(1+(n−1)ρx)Cx2=C12E(ez2)=θ(1+(n−1)ρz)Cz2=C22E(ez2)=θ(1+(n−1)ρz)Cz2=C22E(eyex)=θ(1+(n−1)ρy)1/2(1+(n−1)ρx)1/2ρyxCyCx=C0C1E(eyez)=θ(1+(n−1)ρy)1/2(1+(n−1)ρz)1/2ρyzCyCz=C0C2E(exez)=θ(1+(n−1)ρx)1/2(1+(n−1)ρz)1/2ρxzCxCz=C1C2ρyx*=(1+(n−1)ρy)(1+(n−1)ρy),ρyz*=(1+(n−1)ρy)(1+(n−1)ρz) and ρxz*=(1+(n−1)ρx)(1+(n−1)ρz)ρy=E(yij−Y¯) (yij′−Y¯)E(yij−Y¯)2,ρx=E(xij−X¯) (xij′−X¯)E(xij−X¯)2and ρz=E(zij−Z¯) (zij′−Z¯)E(zij−Z¯)2θ=(N−1)/Nn, ρij*=(1+(n−1)ρi)/(1+(n−1)ρj)ρij=Sij/SiSj and Hij=ρijCi/Cj where i=x,y,zj=x,y,zi≠j

where ρ_y, ρ_x and ρ_z are the intraclass correlation coefficients for study variable y and both auxiliary variables x and z, respectively and ρ_ij is the correlation coefficient between study variable and auxiliary variables. Also the quantities C_y, C_x and C_z are population coefficients of variation of the study variable and the auxiliary variables respectively.

The unbiased mean estimator, without using auxiliary information, together with the expression for the variance in systematic sampling is defined as:

(2)t0=y¯sys,

(3)var(t0)=Y¯2C02.

Swain (1964) and Shukla (1971) proposed classical ratio and product-type estimators under systematic sampling is given by

(4)t1=y¯sysx¯sysX¯,

(5)t2=y¯sysZ¯z¯sys.

The expressions for the mean square error for the estimators t₁ and t₂ upto the first order of approximation are given respectively by

(6)MSE(t1)=Y¯2(C02+C12(1−2Hyxρyx*)),

(7)MSE(t2)=Y¯2(C02+C22(1+2Hyzρyz*)).

The traditional regression estimator for population mean under systematic sampling is given by

(8)t3=y¯sys+byx (X¯−x¯sys),

where b_yx is the sample regression coefficient between y and x. The expression for the mean square error for the estimator t₃, up to the first order approximation is given as

(9)MSE(t3)=Y¯2C02(1−ρyx2).

Singh et al. (2011) proposed exponential ratio and product-type estimators for finite population mean under systematic sampling. The proposed estimators are given by

(10)t4=y¯sys exp (X¯−x¯sysX¯+x¯sys),

(11)t5=y¯sys exp (z¯sys−Z¯Z¯+z¯sys).

The expressions for mean square error for the estimators t₄ and t₅ using first order approximation given respectively by

(12)MSE(t4)=Y¯2(C02+0.25C12(1−4Hyxρyx*)),

(13)MSE(t5)=Y¯2(C02+0.25C22(1+4Hyzρyz*)).

Tailor et al. (2013) proposed the following ratio-cum-product type estimator for finite population mean under systematic sampling:

(14)t6=y¯sys (X¯x¯sys) (z¯sysZ¯).

The expression for the mean square error of estimator t₆, up to first order approximation, is given by:

(15)MSE(t6)=Y¯2(C02+C12(1−2Hyxρyx*)+C22(1+2Hyzρyz*−2Hxzρxz*)).

Khan (2016) proposed a generalized class of exponential estimators for the estimation of finite population mean under systematic sampling. The estimator with the expression for the mean square error is given by:

(16)t7=y¯sys exp (αX¯−x¯sysX¯+(a−1)x¯sys+βZ¯−z¯sysZ¯+(b−1)z¯sys),

(17)MSE(t6)=Y¯2(C02+JC12(J−2Hyxρyx*)+KC22(K−2Hyzρyz*+2JHxzρxz*)),

where

a=αJ,J=ρyx*Hyx−HyzHzx(1−ρxz2),

and

b=βM,M=ρyz*Hyz−HyxHxz(1−ρxz2).

3. Generalized Semi-Exponential Type Estimator

In this section, a generalized semi-exponential type mean estimator is proposed making use of two auxiliary variables under the framework of systematic sampling. The proposed estimator is:

(18)tGE=y¯sys(X¯x¯sys)v1 exp (αZ¯−z¯sysZ¯+(v2−1)z¯sys),

where v₁ and v₂ (v₂> 0) are constants that need to be optimized and estimated for the expression of minimum value of the mean square error of the proposed estimator t_GE. A generalized constant α can assume values -1, 0 and 1 to give various special cases of the proposed estimators.

In order to obtain the expression of mean square error of the proposed estimator, we expand the proposed estimator expression using the notations given in (1), and get.

(19)tGE=Y¯(1+ey)(X¯X¯(1+ex))v1 exp (αZ¯−Z¯−Z¯ezZ¯+Z¯(v2−1)(1+Z¯ez)).

Applying Taylor series and ignoring the terms beyond the second order of approximation and taking expectation on both sides of (19), we get

(20)E(tGE−Y¯)=Y¯E(ey−ex−v1eyex+v12ex2)(1−{−αv2ez+αv2ez2−αv22ez2}+α2v22ez2).

After simplification of (20), we have

(21)bias(tGE)=Y¯(v1C12(1−Hyxρyx*)+α2a2C22{2(a−1)+α−Hyzρyz*−v1Hzxρzx*}).

In order to obtain the expression for the mean square error of the proposed estimator, applying Taylor series and ignoring the terms up to the first order approximation of (19), we have

(22)tGE=Y¯(1+ey−v1ex) exp (−αv2ez).

Squaring and taking expectation on both sides of (22), we get

(23)E(tGE−Y¯)2=Y¯2E(ey−v1ex−αv2ez)2.

The expression for the mean square error of the proposed estimator is given as

(24)MSE(tGE)=Y¯2(C02+v1 C12(v1−2Hyxρyx*)+αv2C22{αv2−2Hyzρyz*+2v1Hxzρxz*}).

The optimum values of v₁ and v₂ are

v1=ρyx*Hyx−HyzHzx(1−ρxz2)=J and v2(opt)=αM,

where

M=ρyz*Hyz−HyxHxz(1−ρxz2).

The expression of minimum mean square error of the proposed estimator t_GE is given as:

(25)MSEmin(tsysGE)=Y¯2(C02+JC12(J−2Hyxρyx*)+MC22{M−2Hyzρyz*+2JHxzρxz*}).

It is noticed that for different values of v₁, v₂ and α we may get various forms of exponential and semi exponential type estimators as new families of t_GE, as given in Table 1 (Appendix A).

4. Relative Performance of Proposed Estimator Compared to other Estimators

In this section, the theoretical comparisons of the proposed estimator are given with some relevant competing estimators.

i.
The proposed generalized semi exponential type estimator t_GE will be more precise estimator then the unbiased mean per unit estimator given in (3) when
MSEmin(tGE)≤Var(t0), Y¯2(C02+JC12(J−2Hyxρyx*)+MC22(M−2Hyzρyz*+2JHxzρxz*))≤Y¯2C02,MC22(2Hyzρyz*−M−2JHxzρxz*)−JC12(J−2Hyxρyx*)≥0.
ii.
The proposed estimator t_GE will be more precise estimator than the classical ratio estimator given in (6) when
MSEmin(tGE)≤MSE(t1),Y¯2(C02+JC12(J−2Hyxρyx*)+MC22{M−2Hyzρyz*+2JHxzρxz*})≤Y¯2(C02+C12(1−2Hyxρyx*)),C12(1−2Hyxρyx*)−JC12(J−2Hyxρyx*)−MC22(M−2Hyzρyz*+2JHxzρxz*)≥0.
iii.
The proposed estimator t_GE will be more precise estimator than the product estimator given in (7) when
MSEmin(tGE)≤MSE(t2).AC12(A−2Hyxρyx*)+MC22(M−2Hyzρyz*+2AHxzρxz*)≤C22(1+2Hyzρyz*),C22(1+2Hyzρyz*)−AC12(A−2Hyxρyx*)−MC22(M−2Hyzρyz*+2AHxzρxz*)≥0.
iv.
The proposed estimator t_GE will be more precise estimator than the regression estimator given in (9) when
MSEmin(tGE)≤MSE(t3), Y¯2(C02+JC12(J−2Hyxρyx*)+MC22(M−2Hyzρyz*+2JHxzρxz*))≤Y¯2C02(1−ρyx2),ρyx2≥(JC12(2Hyxρyx*−J)+MC22(2Hyzρyz*−M−2JHxzρxz*))/C02.
v.
The proposed estimator t_GE will be more precise estimator than the exponential ratio type estimator given in (12) when
MSEmin(tGE)≤MSE(t4),Y¯2(C02+JC12(J−2Hyxρyx*)+MC22(M−2Hyzρyz*+2JHxzρxz*))≤Y¯2(C02+0.25C12(1−4Hyxρyx*)),C12(0.25−4Hyxρyx*)−JC12(J−2Hyxρyx*)−MC22(M−2Hyzρyz*+2JHxzρxz*)≥0.
vi.
The proposed estimator t_GE will be more precise estimator than the exponential product-type estimator given in (13) when
MSEmin(tGE)≤MSE(t5),Y¯2(C02+JC12(J−2Hyxρyx*)+MC22(M−2Hyzρyz*+2JHxzρxz*))≤Y¯2(C02+0.25C22(1−4Hyzρyz*)), C22(0.25−4Hyzρyz*)−JC12(J−2Hyxρyx*) −MC22(M−2Hyzρyz*+2JHxzρxz*)≥0.
vii.
The proposed estimator t_GE will be more precise estimator than the ratio-cum-product type estimator given in (15) when
MSEmin(tGE)≤MSE(t6),Y¯2(C02+JC12(J−2Hyxρyx*)+MC22(M−2Hyzρyz*+2JHxzρxz*))≤Y¯2(C02+C12(1−2Hyxρyx*)+C22(1+2Hyzρyz*−2Hxzρxz*)), C12(1−2Hyxρyx*)+C22(1+2Hyzρyz*−2Hxzρxz*)−JC12 (J−2Hyxρyx*)−MC22(M−2Hyzρyz*+2JHxzρxz*)≥0.
viii.
The proposed estimator t_GE will be more precise estimator than the generalized exponential-cum-exponential type estimator given in (17) when
MSEmin(tGE)≤MSE(t7),Y¯2(C02+JC12(J−2Hyxρyx*)+MC22(M−2Hyzρyz*+2JHxzρxz*))≤Y¯2(C02+JC12(J−2Hyxρyx*)+KC22(K−2Hyzρyz*+2JHxzρxz*)), C12(J−Hyxρyx*)+MC22Hxzρxz*≥0.

5. Empirical Study

Evaluation of the proposed estimator is based on the percentage relative efficiencies (PRE’s) compared to the traditional unbiased mean estimator. So the value greater than one hundred indicates that the competing estimators are more efficient than the usual mean estimator. The PRE’s of all the estimators over the mean per unit estimator can be obtained from the following mathematical formula:

PRE=var(t0)MSE (ti)×100, i=0,1,2,…,7,G1,G2,…,G10.

Here, we consider a population data set from the literature given in Table 2, to examine the performance of the proposed estimator over the other competing estimators at optimum conditions.

N	15	n	3	-	-
Y¯	80	C_y	0.56	ρ_yx	0.9848
X¯	44.47	C_x	0.28	ρ_yz	-0.9760
Z¯	48.40	C_z	0.43	ρ_xz	-0.9539
S_yx	538.57	Sy2	2000	ρ_y	0.6652
S_yz	-902.87	Sx2	149.55	ρ_x	0.7070
S_xz	-241.06	Sz2	427.83	ρ_z	0.5487

Table2:

Population (Source: Tailor et al. (2013)

The results of MSE’s and PRE’s of all the estimators considered in this paper are summarized in Table 3.

*t_i*	*MSE*	*PRE’s*	*t_i*	*MSE*	*PRE’s*
t₀	1344.07	100.00	t_G1	4408.34	33.01
t₁	373.32	389.62	t_G2	46.27	3144.94
t₂	768.06	189.45	t_G3	2249.14	64.70
t₃	43.74	3362.67	t_G4	1131.00	128.66
t₄	820.09	177.43	t_G5	93.11	1562.88
t₅	1044.42	139.32	t_G6	52.99	2746.27
t₆	187.08	777.79	t_G7	7321.20	19.88
t₇	23.67	6158.08	t₇₈	5100.04	28.53
*t_GE*	*21.62*	*6729.21*	t_G9	753.10	193.22
			t_G10	1816.00	80.13

Table 3:

MSE’s and PRE’s of all the estimators

The results presented in Table 3 indicate that the proposed generalized semi-exponential type estimator works considerably better than all other exponential and non-exponential estimators considered in this paper. The proposed estimator has the least mean square error as compare to other estimators.

6. Conclusion

A generalized semi-exponential type estimator is proposed in this paper for the estimation of finite population mean with two auxiliary variables under the framework of systematic sampling. The expressions of approximate bias and mean square error of the proposed estimator are derived. The algebraic expressions for the mean square error of the proposed estimator are compared with other existing estimators both theoretically and empirically. The efficiency comparisons are also carried out using the data taken from Tailor et al. (2013). The results illustrated in Table 3 show that the proposed generalized semi-exponential type estimator is more efficient than the other estimators considered in this paper in term of higher percent relative efficiency.

Acknowledgement

Authors are thankful to the editor-in-chief and the learned referee for their inspiring and fruitful suggestions

APPENDIX A

*Estimators*	*Mean Square Errors*	v₁	α	v₂
t0=y¯sys	MSE(t0)=Y¯2C02	0	0	0
tG1=y¯sys exp (Z¯−z¯sysZ¯)	MSE(tG1)=Y¯2(C02+C22(1−2Hyzρyz*))	0	1	1
t4=y¯sys exp (Z¯−z¯sysZ¯+z¯sys)	MSE(t4)=Y¯2(C02+0.25C22(1−4Hyzρyz*))	0	1	2
tG2=y¯sys exp (z¯sys−Z¯Z¯)	MSE(tG2)=Y¯2(C02+C22(1+2Hyzρyz*))	0	-1	1
t5=y¯sys exp (z¯sys−Z¯Z¯+z¯sys)	MSE(t5)=Y¯2(C02+0.25C22(1+4Hyzρyz*))	0	-1	2
t1=y¯sys(X¯x¯sys)	MSE(t1)=Y¯2(C02+C12(1−2Hyxρyx*))	1	0	0
tG3=y¯sys(X¯x¯sys) exp (Z¯−z¯sysZ¯)	MSE(tG3)=Y¯2(C02+C12(1−2Hyxρyx)+C22(1−2Hyzρyz+2Hxzρxz*))	1	1	1
tG4=y¯sys(X¯x¯sys) exp (Z¯−z¯sysZ¯+z¯sys)	MSE(tG4)=Y¯2(C02+C12(1−2Hyxρyx)+0.25C22(1−4Hyzρyz+4Hxzρxz*))	1	1	2
tG5=y¯sys(X¯x¯sys) exp (z¯sys−Z¯Z¯)	MSE(tG5)=Y¯2(C02+C12(1−2Hyxρyx)+C22(1+2Hyzρyz−2Hxzρxz*))	1	-1	1
tG6=y¯sys(X¯x¯sys) exp (z¯sys−Z¯Z¯+z¯sys)	MSE(tG6)=Y¯2(C02+C12(1−2Hyxρyx)+0.25C22(1+4Hyzρyz−4Hxzρxz*))	1	-1	2
t2=y¯sys(x¯sysX¯)	MSE(t1)=Y¯2(C02+C12(1+2Hyxρyx*))	-1	0	0
tG7=y¯sys(x¯sysX¯) exp (Z¯−z¯sysZ¯)	MSE(tG7)=Y¯2(C02+C12(1+2Hyxρyx)+C22(1−2Hyzρyz−2Hxzρxz*))	-1	1	1
tG8=y¯sys(x¯sysX¯) exp (Z¯−z¯sysZ¯+z¯sys)	MSE(tG8)=Y¯2(C02+C12(1+2Hyxρyx)+0.25C22(1−4Hyzρyz−4Hxzρxz*))	-1	1	2
tG9=y¯sys(x¯sysX¯) exp (z¯sys−Z¯Z¯)	MSE(tG9)=Y¯2(C02+C12(1+2Hyxρyx)+C22(1+2Hyzρyz+2Hxzρxz*))	-1	-1	1
tG10=y¯sys(x¯sysX¯) exp (z¯sys−Z¯Z¯+z¯sys)	MSE(tG10)=Y¯2(C02+C12(1+2Hyxρyx)+0.25C22(1+4Hyzρyz+4Hxzρxz*))	-1	-1	2

Table 1:

Some Special Cases of Proposed Estimator

References

1.AKPC Swain, “The use of systematic sampling ratio estimate”, Journal of the Indian Statistical Association, Vol. 2, 1964, pp. 160-164.

2.Banarasi, SNS Kushwaha, and KS Kushwaha, A class of ratio, product and difference type estimators in systematic sampling, Microelectronics Reliability, Vol. 33, No. 4, 1993, pp. 455-457.

3.HP Singh and R Singh, Almost unbiased ratio and product type estimators in systematic sampling, Questiio, Vol. 22, No. 3, 1998, pp. 403-416.

4.HP Singh, R Tailor, and NK Jatwa, Modified ratio and product estimators for population mean in systematic sampling, Journal of Modern Applied Statistical Methods, Vol. 10, No. 2, 2011, pp. 424-435.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 17 - 2
Pages: 283 - 290
Publication Date: 2018/06/30
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.2018.17.2.8 How to use a DOI?
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TY  - JOUR
AU  - Muhammad Nouman Qureshi
AU  - Sadia Khalil
AU  - Muhammad Hanif
PY  - 2018
DA  - 2018/06/30
TI  - Generalized Semi Exponential Type Estimator under Systematic Sampling
JO  - Journal of Statistical Theory and Applications
SP  - 283
EP  - 290
VL  - 17
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.2.8
DO  - 10.2991/jsta.2018.17.2.8
ID  - Qureshi2018
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