Review Article Volume 4 Issue 6
Department of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, India
Correspondence: Jambulingam Subramani, Department of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, RV Nagar, Kalapet, Pondicherry, India
Received: September 26, 2016 | Published: November 15, 2016
Citation: Subramani J, Ajith MS. Modified ratio cum product estimators for estimation of finite population mean with known correlation coefficient. Biom Biostat Int J. 2016;4(6):251-256. DOI: 10.15406/bbij.2016.04.00113
In this paper, a modified ratio cum product estimator for the estimation of finite population mean of the study variable using the known correlation coefficient of the auxiliary variable is introduced. The bias and mean squared error of the proposed estimator are also obtained. The relative performance of the proposed estimator along with some existing estimators is accessed for certain labeled and natural populations. The results show that the proposed estimator is to be more efficient than the existing estimators.
Keywords: bias, mean squared error, natural population, simple random sampling, linear regression estimator
In sampling theory, a wide variety of techniques is used to obtain efficient estimators for the population mean. The commonly used method to obtain the estimator for population mean is simple random sampling without replacement (SRSWOR) when there is no auxiliary variable available. There are methods that use the auxiliary information of the study characteristics. If there exists an auxiliary variable X which is correlated with the study variable Y, then a number of estimators such as ratio, product, modified ratio, modified product, regression estimators and their modifications are widely available for estimation of population mean of the study variable Y.
Consider a finite population U={U1,U2,U3....UN}U={U1,U2,U3....UN} of N distinct and identifiable units. Let Y be the study variable which takes the values Y={Y1,Y2,Y3,...YN}Y={Y1,Y2,Y3,...YN} . Here the problem is to estimate the population mean ˉY=1N∑Ni=1Yi¯¯¯Y=1N∑Ni=1Yi on the basis of a random sample selected from the population U.
Before discussing further the various estimators, the notations to be used in this article are listed here.
N - Population size
n - Sample size
f=nNf=nN
- Sampling fraction
Y - Study variable
X - Auxiliary variable
ˉX,ˉY¯¯¯X,¯¯¯Y
- Population means
ˉx,ˉy¯x,¯y
- Sample means
Sx,SySx,Sy
- Population standard deviations
Sx,SySx,Sy
- Sample standard deviations
Cx,CyCx,Cy
- Coefficient of variations
ρρ
- Correlation coefficient between x and y
β1β1
- Coefficient of skewness
β2β2
- Coefficient of kurtosis
B(.) - Bias of estimators
MSE(.) - Mean squared error of estimators
In simple random sampling without replacement, the estimator ˉysrs¯ysrs is an unbiased estimator for the population mean ˉY¯¯¯Y and its variance is given by
V(ˉysrs)=δS2yV(¯ysrs)=δS2y (1)
Where δ=(1−fn)δ=(1−fn)
Cochran,1 use auxiliary information for the estimation of population mean of the variable under study and proposed the ratio estimator of the population mean ˉY¯¯¯Y of the study variable,
ˆYR=ˉyˉxˉX=ˆRˉXˆYR=¯y¯x¯¯¯X=ˆR¯¯¯X
The bias and mean squared error of the ratio estimator are given by
B(⌢ˉYR)=δˉY[C2x−ρCxCy]B(⌢¯¯¯YR)=δ¯¯¯Y[C2x−ρCxCy]
MSE(⌢ˉYR)=δˉY2[C2y+C2x−2ρCxCy]MSE(⌢¯¯¯YR)=δ¯¯¯Y2[C2y+C2x−2ρCxCy] (2)
The linear regression estimator and its variance are given by
ˉylr=ˉy+b(ˉX−ˉx)¯ylr=¯y+b(¯¯¯X−¯x)
V(ˉylr)=δS2y(1−ρ2)V(¯ylr)=δS2y(1−ρ2) (3)
where b is the regression coefficient Y on X
Murthy2 proposed the product estimator to estimate the population mean of the study variable when there is a negative correlation between the study variable Y and auxiliary variable X as
⌢ˉYp=ˉyˉxˉX⌢¯¯¯Yp=¯y¯x¯¯¯X
The bias and the mean squared error of the product estimator are given by
B(ˉYp)=δˉY[ρCxCy]B(¯¯¯Yp)=δ¯¯¯Y[ρCxCy]
MSE(⌢ˉYp)=δˉY2[C2y+C2x+2ρCxCy]MSE(⌢¯¯¯Yp)=δ¯¯¯Y2[C2y+C2x+2ρCxCy] (4)
Singh and Tailor3 introduced the modified ratio estimator for the population mean with known population correlation coefficient ρ of the auxiliary variable and is given by
⌢ˉYMR=ˉy(ˉX+ρˉx+ρ)⌢¯¯¯YMR=¯y(¯¯¯X+ρ¯x+ρ)
The bias and mean squared error of this modified ratio estimator are given by
B(⌢ˉYMR)=δˉY[θ2C2x−θρCxCy]B(⌢¯¯¯YMR)=δ¯¯¯Y[θ2C2x−θρCxCy]
MSE(⌢ˉYMR)=δˉY2[C2y+θ2C2x−2θρCxCy]MSE(⌢¯¯¯YMR)=δ¯¯¯Y2[C2y+θ2C2x−2θρCxCy]
where θ=↼X↼X+ρ
The modified product estimator with known correlation coefficient of the auxiliary variable when there is a negative correlation between the study variable Y and auxiliary variable X is given as
ˆ↼YMp=↼y(↼x+ρ↼X+ρ)
The bias and mean squared error of the modified product estimator are given by
B (ˆ↼YMp)=δ↼Y[θρCxCy]
MSE(ˆ↼YMp)=δˆ↼Y[C2y+θ2C2x+2θρCxCy] (6)
where θ=↼X↼X+ρ
In literature, several estimators are available with auxiliary variables. However the problem is that the best estimator in terms of bias and efficiency are not fully addressed. In this paper, we attempt to solve such type of problems. The existing estimators are biased but the percentage relative efficiency is better than that of simple random sampling, ratio and product estimators. These points are motivated us to introduce a new class of improved ratio cum product estimators for the estimation of the population mean of the study variable.
For estimating population mean ˉY we have proposed a class of ratio cum product estimators4 for the population mean by using the known population correlation coefficient of the auxiliary variable and is given by
ˆ↼Ypr=αλ1↼y(↼X+ρ↼x+ρ)+(1−α)λ2↼y(↼x+ρ↼X+ρ) (9)
Here, λ1=SySy+γ1Cy and λ2=SySy+γ2Cy , γ1=B(ˆ↼YMR), γ2=B(ˆ↼YMp)
Bias and mean squared error of the proposed estimators
The detailed derivation of the bias and mean squared error are given in the appendix whereas the procedures to obtain the bias and mean squared error of the proposed estimators are briefly outlined below:
Consider eθ=↼y−↼Y↼Y,e1=↼x−↼X↼X , θ=↼X↼X+ρ
E(eθ)=E(e1)=0 , E(e02)=δ↼Y2C2y , E(e12) = δ↼X2C2x, E(e0e1)= δρCxCy
Substitute these values in equation (9) and neglecting the high order expressions, we get
B(ˆ↼YPr)=E(ˆ↼YPr−↼Y)
B(ˆ↼YPr)=↼Y(αλ1+(1−α)λ2−1)+δ↼Y{αλ1θ2C2x−θρCxCy(αλ1−(1−α)λ2)}
MSE(ˆ↼YPr)=↼Y2(A−1)2+δ↼Y2{Cy2(αλ1+(1−α)λ2)2+θ2Cx2(3α2λ12+(1+α)2λ22−2αλ1)+2θρCxCy(αλ1−(1−α)λ2)−2(α2λ12−(1−α)2λ22)}
MSE(ˆ↼YPr)=↼Y2(A−1)2+δ↼Y2{A2Cy2+θ2Cx2(A2+(A+B)(B−1))−2θρCxCyB(2A−1)}
where A=(αλ1+(1−α)λ2) , B=(αλ1−(1−α)λ2) and θ=↼X↼X+ρ
The optimal value of α is determined by minimizing the MSE ( ˆ↼YPr with respect to α. For this differentiate MSE with respect to α and equate to zero.5
∂MSE∂α=0 , and we get the value of α, as
α=(λ2−1)(λ2−λ1)−δ[Cy2λ2(λ1−λ2)−θ2Cx2(λ1+λ22)+θρCxCy(λ1+λ2−4λ22)](λ1−λ2)2+δ[(λ1−λ2)2Cy2+θ2Cx2(3λ12+λ22)4θρCxCy(λ22−λ12)]
Efficiency comparison
The efficiencies of the proposed estimators with that of the existing estimators are obtained algebraically and are as follows:
Comparison of proposed estimator and simple random sampling (SRSWOR) estimator
The proposed estimator is more efficient than simple random sampling estimator,
V(↼ylr)≥MSE(ˆ↼YPr) if
C2y≥{(A−1)2+δ2{θ2C2x(A2+(A+B)(B−1))−2θρCxCyB(2A−1)}}δ2(1−A2)
Comparison of proposed estimator and linear regression estimator
The proposed estimator is more efficient than linear regression estimator,
V(↼ylr)≥MSE(ˆ↼YPr) if
C2y≥{(A−1)2+δ[θ2Cx2(A2+(A+B)(B−1))−2θCxCy(B(2A−1)−1)]}δ(1−ρ2−A2)
Comparison of proposed estimator and ratio estimator
The proposed estimator is more efficient than ratio estimator
MSE(ˆ↼YP)≥MSE(ˆ↼YPr) if
C2y≥{(A−1)2+δ{C2x(θ2(A2+(A+B)(B−1))−1)−2ρCxCy(θB(2A−1)−1)+1}}δ(1−A2)
Comparison of proposed estimator and product estimator
The proposed estimator is more efficient than ratio estimator,6
MSE(ˆ↼YP)≥MSE(ˆ↼YPr) if
C2y≥{(A−1)2+δ{C2x(θ2(A2+(A+B)(B−1))−1)−2ρCxCy(θB(2A−1)+1)}}δ(1−A2)
Comparison of proposed estimator and modified ratio estimator
The proposed estimator is more efficient than modified ratio estimator,7
MSE(ˆ↼YMR)≥MSE(ˆ↼YPr)
C2y≥{(A−1)2+δ{θ2C2x(A2+(A+B)(B−1)−1)−2θρCxCy(θB(A−1)−1)−1}}δ(1−A2)
Comparison of proposed estimator and modified product estimator
The proposed estimator is more efficient than modified product estimator,
MSE(ˆ↼YMp)≥MSE(ˆ↼YPr) if
C2y≥{(A−1)2+δ{θ2C2x(A2+(A+B)(B−1)−1)−2θρCxCy(B(2A−1)−1)}}δ(1−A2)
Numerical study
In this section, we consider the four natural populations population 1 Khoshnevisan et al.,8 Population 2 Cochran<>9] (page 325) population 3 and 4 Singh and Chaudhary,10 (page 177) and are used to compare the percentage relative efficiency of proposed estimator with that of the existing estimators such as SRSWOR sample mean, linear regression estimator, ratio estimator, product estimator, modified ratio estimators, and modified product estimators.
We have proposed a class of modified ratio cum product estimators for finite population11 mean of the study variable Y with known correlation coefficient of the auxiliary variable X. The bias and mean squared error of the proposed estimators are obtained and compared with that of the simple random sampling without replacement, regression, ratio, product, modified ratio, modified product estimators by both algebraically and numerically. We support this theoretical result with numerical examples. We have shown that the proposed estimator is more efficient than other existing estimators under the optimum values of α. Table 1&2 shows that the bias and MSE of the proposed estimators are smaller than the other competing estimators. Table 3 shows that the percentage relative efficiency of the proposed estimator with respect to the existing estimators,
Parameters |
Population 1 |
Population 2 |
Population 3 |
N |
20 |
10 |
34 |
n |
8 |
3 |
3 |
ˉY |
19.55 |
101.1 |
856.4117 |
ˉX |
18.8 |
58.8 |
208.8823 |
ρ |
-0.9199 |
0.6515 |
0.4491 |
Sy |
6.9441 |
15.4448 |
733.1407 |
Cy |
0.3552 |
0.1527 |
0.8561 |
Sx |
7.4128 |
7.9414 |
150.5059 |
Cx |
0.3943 |
0.1351 |
0.7205 |
β1 |
3.0613 |
0.2363 |
2.9123 |
β2 |
0.5473 |
2.2388 |
0.9781 |
θ |
1.0514 |
0.989 |
0.9978 |
γ1 |
0.4506 |
0.1072 |
625915 |
γ2 |
-0.1986 |
0.3136 |
71.947 |
λ1 |
0.9774 |
0.9989 |
0.9319 |
λ2 |
1.0102 |
0.9969 |
0.9225 |
*α |
0.1055 |
0.8717 |
0.7614 |
Table 1 The computed values of constants and parameters from different populations
Estimator |
Population 1 |
Population 2 |
Population 3 |
Population 4 |
||||
Bias |
MSE |
Bias |
MSE |
Bias |
MSE |
`Bias |
MSE |
|
Proposed |
1.14e-06 |
0.5463 |
1.13e-15 |
31.9319 |
-0.00274 |
109092.8 |
-0.0015 |
66145.84 |
ˉysrs |
- |
3.6166 |
- |
55.6603 |
- |
163356.4 |
- |
91690.37 |
ˉylr |
- |
0.5561 |
- |
32.0343 |
- |
130408.9 |
- |
73197.27 |
⌢ˉYR |
0.4168 |
15.4595 |
0.1132 |
35.0447 |
63.0193 |
155580.6 |
35.3721 |
87325.9 |
⌢ˉYp |
-0.1889 |
0.6869 |
0.3171 |
163.283 |
72.0984 |
402564.2 |
40.4681 |
225955.4 |
⌢ˉYMR |
0.4506 |
16.3099 |
0.1072 |
34.7991 |
62.5915 |
155359.1 |
35.1319 |
87201.54 |
⌢ˉYMP |
-0.1986 |
0.7774 |
0.3163 |
161.6321 |
71.947 |
401824.2 |
40.3832 |
225540 |
Table 2 Bias and MSE of proposed and Existing Estimators from different population
Estimators |
Population 1 |
Population 2 |
Population 3 |
Population 4 |
ˉysrs |
661.981 |
174.3092 |
149.6356 |
138.6185 |
ˉylr |
101.802 |
100.3205 |
119.4555 |
110.6604 |
⌢ˉYR |
2829.752 |
109.7481 |
142.5216 |
132.0283 |
⌢ˉYp |
125.7468 |
511.3465 |
368.7708 |
341.6196 |
⌢ˉYMR |
2985.408 |
108.979 |
142.183 |
131.8322 |
⌢ˉYMP |
142.2864 |
506.1764 |
367.6544 |
340.9739 |
Table 3 Percentage Relative Efficiency of the Proposed Estimator
In fact, the PRE is ranging from
From this, we have observed that the proposed estimator is performed better than that of other existing estimators and hence we recommend the proposed estimators for the practical problems.
None.
None.
©2016 Subramani, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.
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