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Robust Ratio Estimators Compare with Regression Estimators with Several Auxiliary Variables

Ahmed Audua, Ishaq O. Olawoyinb, Jamiu O. Muilia

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aDepartment of Mathematics, Usmanu Danfidiyo University, Sokoto, Nigeria.

e-mail: [email protected] of Statistics, University of Science and Technology, Wudil, Nigeria.

e-mail: [email protected]

In this paper, the efficiency of estimators with several independent auxiliary variables has been examined in robust regression methods. The mean squared errors of the proposed estimators were derived up to second degree approximation using Taylor’s series approximation.

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The empirical study was conducted and the results revealed that proposed estimators were more efficient.

Keywords: Estimators, Auxiliary variables, Multiple Regression, Outliers, Efficiency

Introduction

The auxiliary variables that are related to the study variables are useful in the improving efficiency of ratio, product and regression estimators in both the planning and estimation stages. However, the efficiency of these aforesaid estimators be affected the existence of extreme values or leverages in the data. Authors such as [1], [2], [3], e.

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t.c. utilized Huber-M estimators in place of least squares estimators to reduce the effects of outliers on the efficiency of [4] estimators. However, case studies in which the variables of the study are associated with independent several auxiliary variables such as expenditure with salaries and teacher/students ratio, GDP with inflation rate, export rate and import rate, obesity with body weight, height and blood pressure etc in estimates using robust regression methods received no attention. Thus, in this study, modified estimators were proposed using Robust Multiple Regression Methods.

The estimators of population mean proposed by [2] using robust regression methods coefficients are;

(1.1)

(1.2)

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(1.3)

(1.4)

(1.5)

where and are population coefficients of variation, kurtosis and robust regression methods.

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(1.6)

where , , , is the sample size, is the population size,are coefficients of slope obtained from Tukey-M [5], Hampel-M [6], Huber-M [7], LMS [8] and LAD [9] methods,, ,, .

Suggested estimators

Having study the work of [2] the suggested estimators are presented in general form as

(2.1) where and are either population coefficients of variation or kurtosis of independent auxiliary variables , but .

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To obtain the mean squared error of , the error terms and are defined such that the expectations are given as

(2.2)

The MSE of to second degree approximation using Taylor series method is obtained as; (See Appendix A for details of derivation)

(2.3)

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If and , then the proposed estimator becomes;

(2.4)

The MSE of is equal to MSE of but is replaced by 1.

If and , then the proposed estimator becomes;

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(2.5)

The MSE of is equal to MSE of but is replaced by .

If and , then the proposed estimator becomes;

(2.6)

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The MSE of is equal to MSE of but is replaced by .

If and , then the proposed estimator becomes;

(2.7)

The MSE of is equal to MSE of but is replaced by .

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If and , then the proposed estimator becomes;

(2.8)

The MSE of is equal to MSE of but is replaced by.

Numerical illustration

A Simulated study is conducted to evaluate the performance of the proposed estimators in terms of [2] estimators. The steps of the stimulation are as follows;

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Step1: sample of size 30,000 from normal populations is drawn without replacement using simple random sampling scheme as

and

Step2: construct regression models as:

(3.1) where are regression coefficient of Huber-M, Tukey-M, Hampel-M, LTS and LAD robust estimators.

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Step 3: calculate MSE as given below;

(3.2) where is the estimated mean with sample sizes and is the population mean.

Tables 1, 2 and 3 showed MSE of proposed and [2] estimators for sample sizes 20, 50 and 100 respectively. The results of the tables revealed that the proposed estimators had minimum MSE compared to its counterparts in [2] under Huber-M, Tukey-M, Hampel-M, LTS and LAD.

Conclusion

From the empirical results, it was found that the proposed estimators are more efficient than the estimators by [2].

Appendix A

(D1)

Express (D1) in term of and defined in (2.2) , we have

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(D2)

(D3)

where

Simplify (A3) to second degree approximation, we have

(D4)

(D5)

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Simplify (D5), square and take expectation, we have

(D6)

Apply the results of (2.2), we obtain the MSE of to second degree approximation as

(D7) where

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References

[1] C. Kad?lar , M. Candan, and H. ??ng?. Ratio estimators using robust regression. Hacettepe Journal of Mathematics and Statistics36 (2007):181–88.

[2] T. Zaman, H. Bulut. Modified ratio estimators using robust regression methods. Commun. Stat.-Theory Methods (2018).[3] T. Zaman. Improved modified ratio estimators using robust regression methods. Appl. Math. Comput, 348 (2019), 627-631.

[4] C. Kadilar, H. ??ng?. Ratio estimators in simple random sampling. Appl. Math. Comput. 151(2004):893–902.

[5] J. W. Tukey. Exploratorydataanalysis.MA:Addison-Wesley, (1977).

[6] F. R. Hampel. A general qualitative definition of robustness. The Annals of Math. Stat. 42(1971):1887-1896.

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[7] V. J. Yohai. High breakdown-point and high efficiency robust estimates for regression. The Annals of Stat.,15(1987):642–656.

[8] P. J. Rousseeuw, A.M. Leroy. Robust regression and outlier detection. Wiley Series in Probability and Mathematical Statistics. NewYork: Wiley (1987).

[9] H. Nadia, and A. A. Mohammad . Model of robust regression with parametric and nonparametric methods. Mathematical Theory and Modeling 3 (2013): 27-39.