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Ayed AL e'damat, Khalid Ul Islam Rather RT&A, No 3 (74) A NEW EXPONENTIAL TYPE RATIO ESTIMATOR ...._Volume 18, September 2023
A NEW EXPONENTIAL TYPE RATIO ESTIMATOR FOR THE POPULATION MEAN IN SYSTEMATIC
SAMPLING
Ayed AL e'damat1 and Khalid Ul Islam Rather2*
Department of Mathematics, AL Hussain Bin Talal University, ma'an, Jordan 2Division of Statistics and Computer Science, SKUAST-Jammu, India2 ayed.h. [email protected] [email protected]
Abstract
Utilizing auxiliary information effectively in sample surveys can enhance the accuracy of estimations by capitalizing on the relationship between the main variable under study and the auxiliary variable. Estimators such as ratio, product, exponential, and regression estimators are frequently employed either during the estimation process, the design phase, or both. In everyday situations, it is common to incorporate information from one or two auxiliary variables to improve the precision of estimators. Auxiliary information has been in practice in sampling theory since the advent of modern sample surveys. Information on auxiliary variable having high correlation with the variable under study is quite useful in improving the sampling design. Cochran (1940) used the highly positively correlated study and auxiliary variable to propound the ratio estimator. Product estimator requires a high negative correlation between study and auxiliary variable. By reviewing the literature, it is concluded that applying the auxiliary information enhances the efficiencies of the estimators for estimating any parameter under consideration. So it is well established fact that the use of auxiliary variable technique improves the estimation process for target population. It is also noticed that ratio method of estimation is relatively simple and one of the commonly used methods of estimation. Due to limitations in terms of time and cost, sample surveys are often preferred over census surveys for collecting primary data. In these sample surveys, the ratio, product, and regression estimators are frequently employed to estimate the mean or other parameters of interest for the variable under study. To assess their efficiency, these estimators are compared based on their approximate mean squared errors. In this paper we proposed an exponential ratio type estimator for the estimation of finite population mean under systematic sampling. The mean square error of the proposed estimator is computed up to the first order of approximation and we find proposed estimator is efficient as compared to other existing estimators. Furthermore this theoretical result is supported by numerical examples as well.
Keywords: Systematic sampling, exponential ratio type estimator, mean square error, efficiency.
1. Introduction
In the literature of survey sampling, it is well known that the efficiencies of the estimators of the population parameters of the study variable can be increased by the use of auxiliary information related to auxiliary variable x, which is highly correlated with the study variable y. Auxiliary information may be efficiently utilized either at planning stage or at design stage to arrive at an improved estimator compared to those estimators, not utilizing auxiliary information. A simple technique of utilizing the known information of the population parameters of the auxiliary variables is through ratio, product, and regression method of estimations using different probability sampling designs such as simple random sampling, stratified random sampling, cluster sampling, systematic sampling, and double sampling. In the present paper we will use knowledge of the auxiliary variables under the framework of systematic sampling. Due to its simplicity,
systematic sampling is the most commonly used probability design in survey of finite populations; see W. G. Madow and L. H. Madow [23]. Apart from its simplicity, systematic sampling provides estimators which are more efficient than simple random sampling or stratified random Sampling for certain types of population; see Cochran [24], Gautschi [25], and Hajeck [9]. later on the problem of estimating the population mean using information on auxiliary variable has also been discussed by various authors including Quenouille [15], Hansen et al. [12], Swain [1], Singh [14], Shukla [16], Srivastava and Jhajj [21], Kushwaha and Singh [10], Bahl and Tuteja [20], Banarasi et al. [2], H. P. Singh and R. Singh [5], Kadilar and Cingi [3], Koyuncu and Kadilar [17], Singh et al. [8], Singh and Solanki [6], Singh and Jatwa [7], Tailor et al. [22], Khan and Singh [13], and Khan and Abdullah [11], R. Singh et al. [18], R. Singh et al. [19], D. S. Robson [4].
Consider a finite population U = U1,U2,U3........UN of size N units numbered from 1 to N in
some order .A sample of size n is taken size n units is taken at random from the first k units and every kth subsequent unit; then, = nk where n and k are positive integers; thus, there will be k samples (clusters) each of size n and observe the study variate y and auxiliary variate x for each and every unit selected in the sample
Let (YijXij) for i = 1.2.....k, j = 1,2........n denote the value of y'th unit in the ith sample.
Then, the systematic sample means are defined as follows:
yst = t0 = i/nYij=iyij, population means
and xst = t0 = 1/n YIj=1yij, are the unbiased estimators of the
Y = 1/nY]j=1yiji and X = 1/n Yj=1xijjof y on x
To obtain the properties of the estimators up to first order of approximation, we use the following errors terms:
eo = ysys — Y/Y, ei
ei = x.
sys
— X/X , e2 = zsys — Z/Z ,Such that E(e1) = 0 for i = 0,1 and 2
and
-'yx
^yx Sy^x
Pyz
•Jyz
SySz
Pxz = ■
k =
Pyx^y
k* =
Pyz^y
cv
P*y ={1+(n — 1)py} P*X = {1 + (n — 1)px}
P*z = {1 + (n — 1)pz} p*
r v
p" = ■
p*
s
xz
c
X
X
Ayed AL e'damat, Khalid Ul Islam Rather RT&A, No 3 (74) A NEW EXPONENTIAL TYPE RATIO ESTIMATOR .._Volume 18, September 2023
P"
P2" = — p"
P"y
** _ _y
P1 ~P",
Where
py ,px,pz are intra class correlation among the pair of units for the variables y,x and z.
2. Estimators in Literature:
In this part, we consider some estimators of the finite population mean in the sampling literature. The variance and MSE's of all the estimators computed here are obtained in the first order of approximation.
The variance of the unbiased estimator for population mean is
var(t0) = AF>o (1.1)
Swain [14] and Shukla [16] suggested the classical ratio and product estimators for finite population mean by are given by
t^^^ a 2)
t2 = ysyexp (1.3)
The mean square errors of the estimators to the first order of approximation are given as follows:
MSE(t1) = AY2[cpo + cp2(l-2kfir)] (1.4)
MSE(t2) = AY2 foo + ^s(1 + 2 kj^r)] (1.5)
Singh et al. [20] utilizing the known knowledge of the auxiliary variable, suggested the following ratio and product type exponential estimators:
t3 = ^exp(i+f;) (L6)
MSE(t3) = A?2 [po + f(1- 4kfi""j] (1.8)
MSE(t4) = AY2 [cpo + f(1 + 4kfi"")] (1.9)
Tailor et al. [25] define the following ratio-cum product estimator for the population mean Y:
'5-Mm
The mean square error of the estimator t6 up to first order of approximation, is given by MSE(ts) = AY2[9o + cp2(l - + cp3 (1- +
Where, Vo = Py Cy
(20)
(2.1)
<Pi = 2Cx2jPy*Px*
92 = Px*Cx2
<P-3 = P/Cz2
cp4 = k*C2
3. Proposed estimator
In this section, we have proposed an exponential ratio type estimator for population mean of the study variable y under systematic sampling as given by:
(2.2)
The first order of approximation of the above error terms is given by E(eo2) = Apy*Cy2, E(ei2) = PX'CX2
E(eoe1) = AC2xJpy*px
Where, A = (—)
\nN )
Expressing (2.2) in terms of e's
tRK = Y(1 + £o)(1 + £i)aexp ( -=—=
X — X(1 + £1)
X + X(1 + £1)
= Y(1 + £o)(1 + £i)aexp (l + |)
(2.3)
tRK = Y(1 + £o)(1 + £i)aexp
2
fi-f+v
a(a — 1)
tRK = Y(1 + £o) (1 + a£i +---£\ + • ) exp
From (2.4)
tBK — Y = Y
a£i +
1 1 2V1+T + T +
a(a — 1)2 £1 ae^2 3e12 e0e1
(2.4)
„ ■ „ +——+ £0 + ae1£0 + ■ „
2 2 8 0 1 0 2
Squaring (2.5) and then taking expectation on both sides, the MSE of the estimator yRK is
z
1
RK
2
MSE(tRK) = AY2
^ V A- ( 2 i ^2 ,^2
9o ^—¡T + aK<Pi + (a2 + a)Y + ~8
(2.6)
Obtain the optimum a to minimize MSE(yRK) . Differentiating MSE(yRK) w.r.t a and equating the derivative to zero. Optimum value of a is given by:
a = --
(Kepi + (P2) 2<P2
Substituting the value of aopt in (2.6), we get the minimum value of yRK as: MSEmin(tRK) = AY2y0[l - pyx2]
(2.7)
It follows from (2.7) that the proposed estimator yRK at its optimum condition is equal efficient as that of the usual linear regression estimator.
4. Efficiency Comparisons
In this section, the MSE of traditional estimators to,t1, t2, t3, t4 and t5 are compared with the MSE of proposed estimator yRK.
From (1.1) - (2.0) and (2.1)
[var(to) - MSEmin(yRK)] > 0
[A72cpopyx2] > 0 (2.8)
[MSE(ti) - MSEmin(yRK)] > 0
AY2[cpo(1 - 2kfirj] - [AY2cpoPyx2] > 0 (2.9)
[MSE(¡2) - MSEmin(yRK)] > 0
AY2[cpo(1 + 2kfi**j\ - [A?2cpopyx2] > 0 (3.0)
[MSE(t3) - MSEmin(yRK)] > 0
A?2 [^ (1 - 4kfirj] - [AY2yoPyX2] > 0 (3.1)
[MSE(t4) - MSEmin(yRK)] > 0
A?2 [^ (1 + 4kfiT*j] - [A?2<popyx2} > 0 (3.2)
[MSE(ts) - MSEmin(yRK)] > 0 AY2[cpo(1 - 2kfi"") + cp3 (1- 2k"fi1**) + 2cp4fifiT] - [A?2cpopyx2] > 0 (3.3)
5. Empirical Study
To examine the merits of the proposed estimator over the other existing estimators at optimum conditions, we have considered natural population data sets from the literature. The sources of population are given as follows.
(Source: Tailor et al. [20]). Consider Population
N = 15 , n = 3, X = 44.47, Y = 80, Z = 48.40, Cy = 0.56, Cx = 0.28, Cz = 0.43
= 2000,S2 = 149.55,Si = 427.83, Syx = 538.57,Syz = -902.86, Sxz = -241.06,
Ayed AL e'damat, Khalid Ul Islam Rather RT&A, No 3 (74)
A NEW EXPONENTIAL TYPE RATIO ESTIMATOR .._Volume 18, September 2023
pyx = 0.9848, pyz = -0.9760pxz = -0.9530, py = 0.6652, px = 0.707, pz = 0.5487.
In order to find (PREs) of the estimator, we use the following formula and PRE(ta,t0) = MSE(t0)/MSE(ta) x 100, For a = 0,1,2,3,4,5 and RK
Table 1: The percent relative efficiency of different estimators with respect to t0.
Population
Estimator MSE( ta) PRE (ta, to)
o 1455.08 100.00
i 373.32 389.62
2 768.06 189.45
13 820.09 177.43
t4 1044.42 139.32
ts 187.08 777.79
£rk 43.88 3316.04
Fig 1: Estimator V/s MSE( ta)
MSE
Fig 2: Estimator V/s PRE(ta,t0)
6. Conclusion
In this article, an exponential ratio type estimator has been proposed under systematic design. The mathematical form of the estimator has been derived and its condition of efficiencies has been formulated with respect to some existing estimators from literature. The properties of the proposed estimator are obtained up to first order of approximation .it has been seen that the suggested estimator performed better than the existing estimator both theoretically and empirically.
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