1982/12/01 by Subhash C. Narula, John F. Wellington · 2 citations
Mathematics · Decision Sciences · #Advanced Statistical Methods and Models #Grey System Theory Applications #Forecasting Techniques and Applications
paper · doi:10.2307/1402501
openalex publication_date 1982/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
Summary The minimum sum of absolute errors regression is resistant to outliers and error distributions with long tails. With the development of several efficient algorithms, computer codes and some statistical inference procedures, it is fast becoming a viable and desirable alternative to the popular least squares regression. The objective of this paper is to present a state of the art survey for the minimum sum of absolute errors regression. The least squares regression has dominated the statistical literature for a long time. This dominance and popularity of the least squares regression can be ascribed, at least partially, to the fact that the theory is simple, well developed and documented. The computer packages are also easily available. The least squares regression is optimal and results in the maximum likelihood estimators of the unknown parameters of the model if the errors are independent and follow a normal distribution with mean zero and a common (though unknown) variance o2. The least squares regression is very far from the optimal in many non-Gaussian situations, especially when the errors follow distributions with longer tails. For the regression problems, Huber (1973) stated, 'just a single grossly outlying observation may spoil the least squares estimate, and, moreover, outliers are much harder to spot in the regression than in the simple location case'. The outliers occurring with extreme values of the regressor variables can be especially disruptive. Andrews (1974) noted that even when the errors follow a normal distribution, alternatives to least squares may be required; especially if the form of the model is not exactly known. Further, least squares is not very satisfactory if the quadratic loss function is not a satisfactory measure of the loss. Loss denotes the seriousness of the nonzero prediction error to the investigator, where prediction error is the difference between the predicted and the observed value of the response variable. Meyer & Glauber (1964) stated that for at least certain economic problems absolute error may be a more satisfactory measure of loss than the squared error. The minimum sum of absolute errors regression (or for brevity, absolute errors regression) overcomes the aforementioned drawbacks of the least squares regression and provides an attractive alternative. It is less sensitive than least squares regression to the extreme errors and assumes absolute error loss function. Because of its resistance to