1. Introduction

In sampling theory, auxiliary information is used widely at both the stages of selection and estimation. At the estimation stage, auxiliary information is used by formulating various types of estimators of different population parameters with a view of getting increased efficiency and are available in plenty in the literature.

Let U = (1, 2, . . . , N) be a finite population of N units with Y being the study variable taking the value for the unit i of U and X being the auxiliary variable taking the value for the unit i of the population, i = 1, 2, . . . , N.

Let be the population of mean of Y and be the population mean of X. Also, let,and

For a simple random sample of size n drawn from U with the sample observations , , . . . , on and ,, . . . , on , let and be the sample means of y-values and x-values respectively.

2. The Suggested Estimators

We know that the finite population variance of the study variable is

(2.1)

From (2.1) it is natural to get an estimator of if we replace and by their some estimators. In particular if is estimated by and is estimated by , we get the estimator of as follows

(2.2)

and its generalized estimator as

(2.3)

where and and f (u , v) satisfying the validity conditions of Taylor’s series expansion is a bounded function of (u , v) such that f(1, 1) = 1.

Also if is estimated by and is estimated by , we get another estimator of as follows

(2.4)

and its generalized estimator as

(2.5)

where and , f(u) and g (v) both are bounded functions in u and v respectively such that = 1 at the point u = 1 and = 1 at the point v = 1 and both are satisfying the regularity conditions for the validity of Taylor’s series expansion having first two derivatives with respect to u and v respectively to be bounded.

3. Bias and Mean Squared Error of Suggested Estimators

(a) Bias and Mean Square Error of Suggested Estimator

Let , , and ,

For simplicity, it is assumed that the population size N is large enough as compared to the sample size n so that finite population correction terms may be ignored. Now

(3.1)

(3.2)

and

(3.3)

Let us consider the proposed estimator defined in (2.2)

or

(3.4)

Taking expectation on both sides of (3.4), the bias in to the order is given by

Bias

Using values of the expectations given from (3.1) to (3.3), we have

(3.5)

Now squaring (3.4) on both sides and then taking expectation, the mean square error of to the first degree of approximation is given by

or

(3.6)

For minimizing (3.6) in two unknowns and , the two normal equations after differentiating (3.6) partially with respect to and are

(3.7)

(3.8)

Solving (3.7) and (3.8) for and , we get the minimizing optimum values to be

(3.9)

(3.10)

which when substituted in (3.6) gives the minimum value of mean square error of the estimator as

(3.11)

(b) Bias and Mean Square Error of Suggested Estimator

For f₁ and f₂ being the first order partial derivatives of with respect to u and v respectively at the point (1,1), that is

expanding in (2.3) in third order Taylor’s series about point (1, 1), we have

where f₁ and f₂ are already defined; f₁₁, f₂₂ and f₁₂ are the second order partial derivatives given by

and u* = 1 + h ( u- 1) , v *= 1 + h ( v-1) , 0 < h < 1.

or

(3.12)

Taking expectation on both sides of (3.12) and using values of the expectations given from (3.1) to (3.3), the bias in to the order is given by

(3.13)

Now squaring (3.12) on both sides and then taking expectation, the mean square error of to the first degree of approximation is given by

(3.14)

For minimizing (3.14) in two unknowns f₁ and f₂ , the two normal equations after differentiating (3.14) partially with respect to f₁ and f₂ are

(3.15)

(3.16)

Solving (3.15) and (3.16) for f₁ and f₂, we get the minimizing optimum values to be

(3.17)

(3.18)

which when substituted in (3.14) gives the minimum value of mean square error of the estimator as

(3.19)

(c) Bias and Mean Square Error of Suggested Estimator

Let us consider the proposed estimator defined in (2.4)

or

(3.20)

Taking expectation on both sides of (3.20), the bias in up to terms of order is given by

Using values of the expectations given from (3.1) to (3.3), we have

(3.21)

Now squaring (3.20) on both sides and then taking expectation, the mean square error to the first degree of approximation is

or

(3.22)

For minimizing (3.22) in two unknowns and , the two normal equations after differentiating (3.22) partially with respect to and are

(3.23)

(3.24)

Solving (3.23) and (3.24) for and , we get the minimizing optimum values to be

(3.25)

and

(3.26)

which when substituted in (3.22) gives the minimum value of mean square error of the estimator to be

(3.27)

(d) Bias and Mean Square Error of Suggested Estimator

For , and to be first, second and third order derivatives of at the point u = 1, u* = 1 + h₁ (u-1), 0 < h₁ < 1 , also , and to be first, second and third order derivatives of at the point v = 1 and v* = 1 + h₂ (v-1), 0 < h₂ < 1, expanding and in in third order Taylor’s series, we have

or

(3.28)

Taking expectation on both sides of (3.28) and using values of the expectations given from (3.1) to (3.3), the bias in to the order is given by

(3.29)

Now squaring (3.28) on both sides and then taking expectation, the mean square error of to the first degree of approximation is given by

Using values of the expectations given from (3.1) to (3.3), we have

or

(3.30)

For minimizing (3.30) in two unknowns and , the two normal equations after differentiating (3.30) partially with respect to and are

(3.31)

and

(3.32)

Solving (3.31) and (3.32) for and , we get the minimizing optimum values to be

(3.33)

(3.34)

which when substituted in (3.30) gives the minimum value of mean square error as

(3.35)

4. Efficiency Comparison with the Traditional Estimator

As we know that the mean square error of usual conventional unbiased estimator of population variance is and

(4.1)

and

(4.2)

Comparative studies regarding their efficiency over the usual conventional unbiased estimator are carried out with the help of a numerical illustration.

Considering the data given in Cochran (1977) dealing with Paralytic Polio Cases ‘Placebo’ (Y) group, Paralytic Polio Cases in not inoculated group (X), computations of required values of have been done and comparisons are made for a simple random sample of size n. For the data considered, we have

Using above values, we have

Mean Square Error of usual conventional unbiased estimator = 9.534136695 and

Hence the percent relative efficiency (PRE) of the proposed estimators , , , and over the usual conventional estimator are given by

showing that the proposed estimators , , , and are more efficient with highly significant percent relative efficiency over the usual conventional unbiased estimator of the population variance.

5. Conclusions

We have derived new sampling estimators of population variance using auxiliary information in the form of mean and variance both, the bias and mean square error equations are obtained. Using these equations, MSE of proposed estimators are compared with the traditional estimator in theory and shown that the proposed estimators have smaller MSE than the traditional one.

ACKNOWLEDGEMENTS

The authors are thankful to the referees and the Editor- in- chief for providing valuable suggestions regarding improvement of the paper.