Thank you . J F

quantile regression  model  has become increasingly popular since the

seminal work of Koenker and Bassett (1978). In contrast to

the standard regression, quantile regression belongs to a robust model

family, which can give an overall assessment of the predector 

effects at different quantile levels of the response of interest.

Regards,

Rahim

The inefficiency of the ordinary least squares is due to the overdependence on assumptions. It suffers when assumptions are wrong and outliers are present.  There are parametric and non-parametric alternatives. The simplest methods of estimating parameters in a regression model that are less sensitive to outliers than the least squares estimates, is to use least absolute deviations. Even then, gross outliers can still have a considerable impact on the model, motivating research into even more robust approaches.

In 1973, Huber introduced M-estimation for regression (M in M-estimation stands for "maximum likelihood type"). The method is robust to outliers in the response variable, but turned out not to be resistant to outliers in the explanatory variables (leverage points). In fact, when there are outliers in the explanatory variables, the method has no advantage over least squares.

In the 1980s, several alternatives to M-estimation were proposed. A book by Rousseeuw and Leroy discusses such issues. Least trimmed squares (LTS) is a viable alternative and is currently preferred (Rousseeuw and Ryan (1997, 2008)). The Theil–Sen estimator has a lower breakdown point than LTS but is statistically efficient and popular. Another proposed solution was S-estimation. This method finds a line (plane or hyperplane) that minimizes a robust estimate of the scale (from which the method gets the S in its name) of the residuals. This method is highly resistant to leverage points, and is robust to outliers in the response. However, this method was also found to be inefficient.

MM-estimation attempts to retain the robustness and resistance of S-estimation, whilst gaining the efficiency of M-estimation. The method proceeds by finding a highly robust and resistant S-estimate that minimizes an M-estimate of the scale of the residuals (the first M in the method's name). The estimated scale is then held constant whilst a close-by M-estimate of the parameters is located (the second M).

There maybe a learning based form of the ordinary LS.

Important links

http://www.jstor.org/stable/1403077?seq=1#page_scan_tab_contents

http://www.eajournals.org/journals/european-journal-of-statistics-and-probability-ejsp/vol3issue-3september-2015/maximum-likelihood-method-an-alternative-to-ordinary-least-square-method-in-estimating-the-parameters-of-simple-weibull-distribution-using-large-samples-of-type-i-censored-observation/

http://www.desy.de/~blobel/apltalk.pdf

https://www.stat.wisc.edu/courses/st771-newton/papers/green.pdf

http://staff.ulsu.ru/semushin/_index/_pilocus/_gist/docs/mycourseware/2-compumath/2-reading/pdf-s/other_books/Rao%20C.R.,%20H.Toutenberg.%20Linear%20Models..%20Least%20Squares%20and%20Al.pdf

This issue should be of interest to professionals. I fall silent, so as not to interfere with the discussion.

Hi Rahim,

Many thanks for your attention. It is a new information for me but I think it is not deal with Heteroskedasticity problem.

Best Wishes

Huda

Good day for all 

Hi Kandarpa 

I thank your a valuable efforts and the important links. I will read it carefully.

Many thanks for you.

Huda

All the best!

Huda -

Regarding GLS, if you deal with cross sectional data, handling only the naturally occurring heteroscedasticity, with no serial correlation, then weighted least squares (WLS, something of a simplified GLS) regression often performs very well.  The coefficient of heteroscedasticity may be estimated from the data in several ways. Here is at least one of them:    

 https://www.researchgate.net/publication/263809034_Alternative_to_the_Iterated_Reweighted_Least_Squares_Method_-_Apparent_Heteroscedasticity_and_Linear_Regression_Model_Sampling

However, in using this for 'prediction' for establishment surveys, I found it more robust to understate the coefficient of heteroscedasticity, possibly to account for relatively higher measurement error, that is, relatively lower data quality, for smaller 'size' members of the population.

See

 https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys

and

 https://www.researchgate.net/publication/261596397_Ken_Brewer_and_the_coefficient_of_heteroscedasticity_as_used_in_sample_survey_inference

(Note that for multiple regression, the "size" measure used with the coefficient of heteroscedasticity, in the random factor of the estimated residuals, could be a preliminary prediction of y, as opposed to x for the one regressor, linear regression case examined in links above.) 

In another establishment survey context, Anders Holmberg and Bengt Swensson also found it more robust to understate the coefficient of heteroscedasticity:

 Holmberg, A., and Swensson, B. (2001), “On Pareto πps Sampling: Reflections on Unequal

Probability Sampling Strategies,” Theory of Stochastic Processes, Vol. 7, pp. 142-155. 

Royall and Cumberland have provided attempts at more robust variance estimation: 

 Royall, R.M., and W.G. Cumberland (1978), "Variance

Estimation in Finite Population Sampling," JASA, 73, 351-358.

Royall, R.M., and W.G. Cumberland (1981), "An Empirical Study

of the Ratio Estimator and Estimators of Its Variance," JASA, 76,

66-88.

However, it may not only be simpler, but about as robust, or more robust, to understate the coefficient of heteroscedasticity.  For regression to the origin, I have found the classical ratio estimator (CRE, a WLS estimator) to be very robust.  In Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons, pp. 158-160, it can be seen that under conditions I find rather mild, Cochran said that the CRE would be "hard to beat." 

Cheers - Jim 

Article Ken Brewer and the coefficient of heteroscedasticity as used...

Article Properties of Weighted Least Squares Regression for Cutoff S...

Conference Paper Alternative to the Iterated Reweighted Least Squares Method ...

Hi  James ,

I'm so grateful for your attention.

Best Wishes

Huda

Dear Dr Huda

Robust regression is a form of regression analysis designed to circumvent some limitations of traditional parametric and non-parametric methods.

The following paper is very useful

http://www.upa.pdx.edu/IOA/newsom/semclass/ho_estimate.pdf

and

http://e-collection.library.ethz.ch/eserv/eth:22482/eth-22482-01.pdf

Hi Aly,

Thank you very much for your important attached file. I will read it carefully .

Best regards

With my best regards and wishes