A Modified Regression Type Estimator Using Two Auxiliary Variables

Vyas Dubey, Yeesha Verma

School of Studies in Statistics, Pt. Ravishankar Shukla University, Raipur, India

ABSTRACT

In this paper, a modified regression type estimator has been proposed for estimating population mean using two auxiliary variables under simple random sampling. The optimum properties of proposed estimator is determined and we find that the proposed estimator is more efficient than the Desraj (1965) and Srivastva (1967). Empirical studies have also done to demonstrate the efficiency of the proposed estimator.

KEYWORDS: Auxiliary variables, Bias, Mean Square Error, Relative Efficiency, Simple Random Sampling without Replacement (SRSWOR).         

1.      INTRODUCTION

The purpose of the survey sampling is to get accurate information about the population characteristics for improving the efficiency of the estimator under study. The use of the auxiliary information fulfills this purpose. If the study variable ( ) and auxiliary variable ( ) are linearly related Hansen, Hurwitz and Madow (1953) suggested difference method of estimation for estimating population mean.

            Let  and  be sample means of  and respectively based on sample of size n drawn from a population of size N and let  be known population mean of . The difference estimator is define as

                                                                                                                             (1.1)

Where  is constant. Modifying in this estimator Bedi and Hajala (1984) suggested estimator

Where is constant, in general is more efficient than . Further Modifying in difference estimator Dubey and Singh (2001) suggested new estimator

                                                                                         (1.2)

Where and are suitable chosen constants. The estimator  is more efficient than if lies between ( ) i.e.

In many practical situations it is easily see that the study variable  related to two or more auxiliary variables. Let as consider another auxiliary variable . Using two auxiliary variables Desraj (1965) and Srivastva (1967) proposed estimator as

                                                                     (1.3)

                                                                                          (1.4)

where  is sample mean of and is known population mean of .

 and

 ,  ,

Motivated by Dubey and Singh (2001), we proposed a modified regression type estimator using two auxiliary variable which is more efficient than estimator  and estimators.

2.     PROPOSD ESTIMATOR

Let ,  and are constants,  and  are known in advance then we proposed estimator as

                                                                         (2.1)

Let the sample of size n is drawn from the population of size N using simple random sampling without replacement then the Bias and MSE of proposed estimator is given as

                                                                                                        (2.2)

          (2.3)

where , ,

          ,  ,   and

For getting the optimum value of ,  and  for which equation  (2.1)  is minimum , we have 

      ,            ,          

and get the equations

                                                                      (2.4)

                                                                                                      (2.5)

                                                                                                       (2.6)

Solving this equations we have optimum values

                                                                                                                         (2.7)

Where

                                                                                                 (2.8)

                                                                                  (2.9)

Put this optimum values in equation (2.2) and (2.3) we obtain minimum Bias and MSE of the proposed estimator

                                                                                                    (2.10)

                                                                             (2.11)

3.     EFFICIENCY COMPARISION

For comparing efficiency of proposed estimator we have

                                                     (3.1)

From (2.11) and (3.1)

  if                                                                                                                (3.2)

Which is always true.

4.     EMPIRICAL STUDY

For the numerical comparison of the estimators, we using the four natural population data sets.

Population 1 – Source: Cochran (1977)

Y: Number of ‘placebo’ children.

X: Number of paralytic polio cases in the placebo group.

Z: Number of paralytic polio cases in the ‘not inoculated’ group.

Population 2- Source: Murthy (1967)

Y: Area under wheat in 1964.

X: Area under in 1963.

Z: Cultivated area in 1961.

Population 3 – Source: Srivnstava et al. (1989)

Y: The measurement of weight of children (in kg).

X: Mid-arm circumference of children (in cm).

Z: Skull circumference of children (in cm).

Population 4- Source: Sukhatme and Chand (1977)

Y: Apple trees of bearing age in 1964.

X: Bushels of apples harvested in 1964.

Z: Bushels of apples harvested in 1959.

Table – 1

                                      Descriptions of the population parameters

Relative efficiency (RE) of the estimators with respect to which is given in table -2, defined by  for various sample sizes.

                        Relative efficiency of the estimators with respect to

5.     CONCLUSION

From the above table - 2 we can see that the proposed estimator is more efficient than Desraj (1965) and Srivastava (1967) estimators for all the population at various sample sizes. From table -2 (in population 2 and 3) we can also conclude that, when the difference between  and  is minimum the use of proposed estimator is much efficient.

 References

1.     Bedi P.K. and Hajala D. (1984). An estimator for population mean utilizing known coefficient of variation and auxiliary variable. Journal of Statistical Research, 18, 29-33.

2.     Cochran W. G. (1977). Sampling techniques. New-York: John Wiley and Sons

3.     Des Raj (1965). On a Method of Using Multi-Auxiliary Information in Sampling Surveys. Journal of The American Statistical Association, 60(309), 270-277.

4.     Dubey Vyas and. Singh S.K. (2001). An Improved Regression Estimator for Estimating Population Mean. Journal of the Indian Society of Agricultural Statistics, 52(2), 179-183.

5.     Hensen M.N., Hurwitz W.N. and W.G. Madow (1953). Sample Survey Methods and Theory. John Wiley and Sons, New York.

6.     Murty M. N. (1967). Sampling theory and methods. Calcutta, India: Statistical Publishing Society.

7.     Sukhatme B.V.  and Chand L. (1977). Multivariate Ratio-Type estimators. Proceedings of Amarican Statistical Association, Social Statistics Section, 927.931.

8.     Srivanstava R. S., Srivastava S.P. &. Khare B.B. (1989). Chain ratio type estimator for ratio of two population means using auxiliary characters. Communications in statistics- Theory and Methods, 18(9), 3917-3926.

9.     Srivastava S.K. (1967). An Estimator Using Auxiliary Information in Sample Surveys. Calcutta Statistical Association bulleting, 16, 121-132.