Submit manuscript...
eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Review Article Volume 4 Issue 6

Modified ratio cum product estimators for estimation of finite population mean with known correlation coefficient

Jambulingam Subramani, Master Ajith S

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

Download PDF

Abstract

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

Introduction

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=1NNi=1Yi¯¯¯Y=1NNi=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=(1fn)δ=(1fn)

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[C2xρCxCy]B(¯¯¯YR)=δ¯¯¯Y[C2xρCxCy]

MSE(ˉYR)=δˉY2[C2y+C2x2ρCxCy]MSE(¯¯¯YR)=δ¯¯¯Y2[C2y+C2x2ρCxCy]                                     (2)

The linear regression estimator and its variance are given by

ˉylr=ˉy+b(ˉXˉx)¯ylr=¯y+b(¯¯¯X¯x)

V(ˉylr)=δS2y(1ρ2)V(¯ylr)=δS2y(1ρ2)                                                              (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[θ2C2xθρCxCy]B(¯¯¯YMR)=δ¯¯¯Y[θ2C2xθρCxCy]

MSE(ˉYMR)=δˉY2[C2y+θ2C2x2θρCxCy]MSE(¯¯¯YMR)=δ¯¯¯Y2[C2y+θ2C2x2θρCxCy]     

where     θ=XX+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

ˆYMp=y(x+ρX+ρ)   

The bias and mean squared error of the modified product estimator are given by

B (ˆYMp)=δY[θρCxCy]

MSE(ˆYMp)=δˆY[C2y+θ2C2x+2θρCxCy]                                                (6)

where θ=XX+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.

Proposed estimators

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

ˆYpr=αλ1y(X+ρx+ρ)+(1α)λ2y(x+ρX+ρ)                                                                              (9)

Here, λ1=SySy+γ1Cy and λ2=SySy+γ2Cy , γ1=B(ˆYMR), γ2=B(ˆYMp)

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=yYY,e1=xXX , θ=XX+ρ

E(eθ)=E(e1)=0  , E(e02)=δY2C2y , E(e12) = δX2C2x, E(e0e1)= δρCxCy

Substitute these values in equation (9) and neglecting the high order expressions, we get

B(ˆYPr)=E(ˆYPrY)

B(ˆYPr)=Y(αλ1+(1α)λ21)+δY{αλ1θ2C2xθρCxCy(αλ1(1α)λ2)}

MSE(ˆYPr)=Y2(A1)2+δY2{Cy2(αλ1+(1α)λ2)2+θ2Cx2(3α2λ12+(1+α)2λ222αλ1)+2θρCxCy(αλ1(1α)λ2)2(α2λ12(1α)2λ22)}

MSE(ˆYPr)=Y2(A1)2+δY2{A2Cy2+θ2Cx2(A2+(A+B)(B1))2θρCxCyB(2A1)}

where A=(αλ1+(1α)λ2) , B=(αλ1(1α)λ2) and θ=XX+ρ

The optimal value of α is determined by minimizing the MSE ( ˆYPr with respect to α. For this differentiate MSE with respect to α and equate to zero.5

MSEα=0 , and we get the value of α, as

α=(λ21)(λ2λ1)δ[Cy2λ2(λ1λ2)θ2Cx2(λ1+λ22)+θρCxCy(λ1+λ24λ22)](λ1λ2)2+δ[(λ1λ2)2Cy2+θ2Cx2(3λ12+λ22)4θρCxCy(λ22λ12)]

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(ˆYPr)   if

C2y{(A1)2+δ2{θ2C2x(A2+(A+B)(B1))2θρCxCyB(2A1)}}δ2(1A2)

Comparison of proposed estimator and linear regression estimator

The proposed estimator is more efficient than linear regression estimator,

V(ylr)MSE(ˆYPr)   if

C2y{(A1)2+δ[θ2Cx2(A2+(A+B)(B1))2θCxCy(B(2A1)1)]}δ(1ρ2A2)

Comparison of proposed estimator and ratio estimator

The proposed estimator is more efficient than ratio estimator

MSE(ˆYP)MSE(ˆYPr)    if

C2y{(A1)2+δ{C2x(θ2(A2+(A+B)(B1))1)2ρCxCy(θB(2A1)1)+1}}δ(1A2)

Comparison of proposed estimator and product estimator

The proposed estimator is more efficient than ratio estimator,6

MSE(ˆYP)MSE(ˆYPr)   if

C2y{(A1)2+δ{C2x(θ2(A2+(A+B)(B1))1)2ρCxCy(θB(2A1)+1)}}δ(1A2)

Comparison of proposed estimator and modified ratio estimator

The proposed estimator is more efficient than modified ratio estimator,7

MSE(ˆYMR)MSE(ˆYPr)  

C2y{(A1)2+δ{θ2C2x(A2+(A+B)(B1)1)2θρCxCy(θB(A1)1)1}}δ(1A2)

Comparison of proposed estimator and modified product estimator

The proposed estimator is more efficient than modified product estimator,

MSE(ˆYMp)MSE(ˆYPr)    if

C2y{(A1)2+δ{θ2C2x(A2+(A+B)(B1)1)2θρCxCy(B(2A1)1)}}δ(1A2)

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.

Conclusion

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

  1. 138.6185 to 661.9810 in case of SRSWOR sample mean
  2. 100.3205 to 119.4555 in case of Linear Regression Estimator
  3. 109.7481 to 2829.7520 in case of Ratio estimator
  4. 125.7468 to 511.3465 in case of Product estimator
  5. 108.9790 to 2985.4080 in case of Modified Ratio estimator and
  6. 142.2864 to 506.1764 in case of Product estimator

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.

Acknowledgments

None.

Conflicts of interest

None.

References

  1. Cochran WG. The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce. The Journal of Agricultural Science. 1940;30(2):262‒275.
  2. Murthy MN. Product method of estimation. Sankhyā:. The Indian Journal of Statistics. 1964;26(1):69‒74.
  3. Singh HP, Tailor R. Use of known correlation coefficient in estimating the finite population means. Statistics in Transition. 2003;6(4):555‒560.
  4. Ekaette Inyang Enang, Victoria Matthew Akpan, Emmanuel John Ekpenyong. Alternative ratio estimator of population mean in simple random sampling, Journal of Mathematics Research. 2014;6(3).
  5. HousilaP Singh, Surya K Pal, Vishal Mehta. A generalized class of dual to product‒cum‒dual to ratiotype estimators of finite population mean in sample surveys. Appl Math Inf Sci Lett. 2016;4(1):25‒33.
  6. J Subramani G Kumarapandiyan. A class of almost unbiased modified ratio estimators for population mean with known population parameters. Elixir Statistics. 2012;44:7411‒7415.
  7. Murthy MN. Sampling theory and methods. Statistical Publishing Society, Calcutta, India. 1967.
  8. Khoshnevisan M, Singh R, Chauhan P, et al. A general family of estimators for estimating population mean using known value of some population parameter(s). Far East Journal of Theoretical Statistics. 2007;22:181‒191.
  9. Cochran WG. Sampling Techniques. (3rd edn), Wiley Eastern Limited, India. 1977;pp. 448.
  10. Singh D, Chaudhary FS. Theory and analysis of sample survey designs.(1st edn), New Age International Publisher, India, 1986;pp. 332.
  11. Subramani J. "Generalized modified ratio estimator for estimation of finite population mean". Journal of Modern Applied Statistical Methods. 2013;12(2):pp.121‒155.
Creative Commons Attribution License

©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.