Now Accepting Apple Pay

Apple Pay is the easiest and most secure way to pay on StudyMoose in Safari.

Manuscript AIJ2

Categories: Data AnalysisOther


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

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


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.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.

Get quality help now
Marrie pro writer
Verified writer

Proficient in: Data Analysis

5 (204)

“ She followed all my directions. It was really easy to contact her and respond very fast as well. ”

+84 relevant experts are online
Hire writer

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;






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


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,, ,, .

Get to Know The Price Estimate For Your Paper
Number of pages
Email Invalid email

By clicking “Check Writers’ Offers”, you agree to our terms of service and privacy policy. We’ll occasionally send you promo and account related email

"You must agree to out terms of services and privacy policy"
Check writers' offers

You won’t be charged yet!

Suggested estimators

Having studied 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 .

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


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


If and , then the proposed estimator becomes;


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

If and , then the proposed estimator becomes;


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

If and , then the proposed estimator becomes;


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

If and , then the proposed estimator becomes;


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

If and , then the proposed estimator becomes;


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;

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


Step2: construct regression models as:

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

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.


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

Appendix A


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




Simplify (A3) to second degree approximation, we have



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


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

(D7) where


  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.
  3. C. Kadilar, H. ??ng?. Ratio estimators in simple random sampling. Appl. Math. Comput. 151(2004):893-902.
  4. J. W. Tukey. Exploratorydataanalysis.MA:Addison-Wesley, (1977).
  5. F. R. Hampel. A general qualitative definition of robustness. The Annals of Math. Stat. 42(1971):1887-1896.
  6. V. J. Yohai. High breakdown-point and high efficiency robust estimates for regression. The Annals of Stat.,15(1987):642-656.
  7. P. J. Rousseeuw, A.M. Leroy. Robust regression and outlier detection. Wiley Series in Probability and Mathematical Statistics. NewYork: Wiley (1987).
  8. H. Nadia, and A. A. Mohammad . Model of robust regression with parametric and nonparametric methods. Mathematical Theory and Modeling 3 (2013): 27-39.

Cite this page

Manuscript AIJ2. (2019, Nov 20). Retrieved from

👋 Hi! I’m your smart assistant Amy!

Don’t know where to start? Type your requirements and I’ll connect you to an academic expert within 3 minutes.

get help with your assignment