Some robust methods for estimating random linear regression model
Keywords:random explanatory variable; random linear regression; modified maximum likelihood; M method; robust empirical likelihood.
The regression model with a random explanatory variable is one of the widely used models in representing the regression relationship between variables in various economic or life phenomena. This model is called the random linear regression model. In this model, the different values of the explanatory variable occur with a certain probability, instead of being fixed in repeated samples, so it contradicts one of the basic assumptions of the linear regression model, which leads to the least squares estimators losing some or all of their optimal properties, depending on the nature of the relationship between the explanatory variable . random and random error limits.
As an alternative to the ordinary least squares method, the modified maximum likelihood method, which was previously used by many researchers, was used to estimate the coefficients of the random regression model. also, two methods were employed, the M method and the robust empirical likelihood method, which were not used previously in estimating the coefficients of the random regression model, but were used in estimating linear regression models that suffer from some standard problems, or in the event that the sample values contain outliers or extreme values. The three methods are among the robust estimation methods.
For the purpose of comparison between the aforementioned estimation methods, simulated experiments were conducted, which revealed the preference of the modified maximum likelihood method despite the great convergence between the estimation results of the three methods. We also carried out a practical application to estimate the regression relationship between packed cell volume (PCV) as a response variable and random blood sugar (RBS) as a random explanatory variable, based on the data of a random sample of 30 patients with heart disease. The results of the practical application were consistent with the results of the simulation experiments.