In applied statistics regression is beyond doubt one of the most widely applied tool in establishing the relationship between a set of predictors and a response variable. Given that in presence of outliers the Maximum Likelihood estimates (ML) are very unstable, in this paper we investigate on the use of the Minimum Density Power Divergence criterion (Basu, A. Harris, I. R Hjort, N.L. and Jones, M.C., Robust and Efficient Estimation by Minimizing a Density Power Divergence, Biometrika, 1998, 85, 549-559) as practical estimation tool for robust parametric regression models. From a realistic point of view, in the absence of information about the nature of the potential data contamination, the choice of a robust estimator can only be achieved resorting to a data-driven mechanism. In this context we suggest a technique that allows to choose the tuning parameter of the Minimum Density Power Divergence estimator (MDPDE) yielding the best estimated regression model for a given sample of data. This is performed comparing the estimated ML regression model with the ones obtained resorting to of the family of MDPDE. For a given tuning parameter, the comparison relies on the index of similarity between functions proposed by Durio and Isaia (Durio, A. and Isaia, E. D., Clusters Detection in Regression Problems: the Integrated Square Error Approach, 2007, Proceedings of the 12th International Conference on Applied Stochastic Models & Data Analysis) and a Monte Carlo Significance test of hypothesis based on this statistic is introduced in order to verify the hypothesis of similarity between the estimated regression models. The optimal tuning parameter is found by iterating the Monte Carlo Significance test until, for the first time, the hypothesis of similarity between the two estimated regression models is rejected. Theory is outlined and main results of a simulation study, referring to several scenarios featuring common industrial and financial issues, are provided to illustrate and validate the approach we propose.
Choosing a Robust Estimator: A Data Based Procedure
DURIO, Alessandra;ISAIA, Ennio Davide
2008-01-01
Abstract
In applied statistics regression is beyond doubt one of the most widely applied tool in establishing the relationship between a set of predictors and a response variable. Given that in presence of outliers the Maximum Likelihood estimates (ML) are very unstable, in this paper we investigate on the use of the Minimum Density Power Divergence criterion (Basu, A. Harris, I. R Hjort, N.L. and Jones, M.C., Robust and Efficient Estimation by Minimizing a Density Power Divergence, Biometrika, 1998, 85, 549-559) as practical estimation tool for robust parametric regression models. From a realistic point of view, in the absence of information about the nature of the potential data contamination, the choice of a robust estimator can only be achieved resorting to a data-driven mechanism. In this context we suggest a technique that allows to choose the tuning parameter of the Minimum Density Power Divergence estimator (MDPDE) yielding the best estimated regression model for a given sample of data. This is performed comparing the estimated ML regression model with the ones obtained resorting to of the family of MDPDE. For a given tuning parameter, the comparison relies on the index of similarity between functions proposed by Durio and Isaia (Durio, A. and Isaia, E. D., Clusters Detection in Regression Problems: the Integrated Square Error Approach, 2007, Proceedings of the 12th International Conference on Applied Stochastic Models & Data Analysis) and a Monte Carlo Significance test of hypothesis based on this statistic is introduced in order to verify the hypothesis of similarity between the estimated regression models. The optimal tuning parameter is found by iterating the Monte Carlo Significance test until, for the first time, the hypothesis of similarity between the two estimated regression models is rejected. Theory is outlined and main results of a simulation study, referring to several scenarios featuring common industrial and financial issues, are provided to illustrate and validate the approach we propose.
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