Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis
Minimum density power divergence estimation provides a general framework for robust statistics, depending on a parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>, which determines...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-03-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/4/399 |
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| author | Marco Riani Anthony C. Atkinson Aldo Corbellini Domenico Perrotta |
| author_facet | Marco Riani Anthony C. Atkinson Aldo Corbellini Domenico Perrotta |
| author_sort | Marco Riani |
| collection | DOAJ |
| description | Minimum density power divergence estimation provides a general framework for robust statistics, depending on a parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>, which determines the robustness properties of the method. The usual estimation method is numerical minimization of the power divergence. The paper considers the special case of linear regression. We developed an alternative estimation procedure using the methods of S-estimation. The rho function so obtained is proportional to one minus a suitably scaled normal density raised to the power <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>. We used the theory of S-estimation to determine the asymptotic efficiency and breakdown point for this new form of S-estimation. Two sets of comparisons were made. In one, S power divergence is compared with other S-estimators using four distinct rho functions. Plots of efficiency against breakdown point show that the properties of S power divergence are close to those of Tukey’s biweight. The second set of comparisons is between S power divergence estimation and numerical minimization. Monitoring these two procedures in terms of breakdown point shows that the numerical minimization yields a procedure with larger robust residuals and a lower empirical breakdown point, thus providing an estimate of <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> leading to more efficient parameter estimates. |
| first_indexed | 2024-03-10T20:46:24Z |
| format | Article |
| id | doaj.art-ed1afc9e43194067803129c435ee43f0 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T20:46:24Z |
| publishDate | 2020-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-ed1afc9e43194067803129c435ee43f02023-11-19T20:17:27ZengMDPI AGEntropy1099-43002020-03-0122439910.3390/e22040399Robust Regression with Density Power Divergence: Theory, Comparisons, and Data AnalysisMarco Riani0Anthony C. Atkinson1Aldo Corbellini2Domenico Perrotta3Dipartimento di Scienze Economiche e Aziendale and Interdepartmental Centre for Robust Statistics, Università di Parma, l43125 Parma, ItalyThe London School of Economics, London WC2A 2AE, UKDipartimento di Scienze Economiche e Aziendale and Interdepartmental Centre for Robust Statistics, Università di Parma, l43125 Parma, ItalyEuropean Commission, Joint Research Centre, 21027 Ispra, ItalyMinimum density power divergence estimation provides a general framework for robust statistics, depending on a parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>, which determines the robustness properties of the method. The usual estimation method is numerical minimization of the power divergence. The paper considers the special case of linear regression. We developed an alternative estimation procedure using the methods of S-estimation. The rho function so obtained is proportional to one minus a suitably scaled normal density raised to the power <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>. We used the theory of S-estimation to determine the asymptotic efficiency and breakdown point for this new form of S-estimation. Two sets of comparisons were made. In one, S power divergence is compared with other S-estimators using four distinct rho functions. Plots of efficiency against breakdown point show that the properties of S power divergence are close to those of Tukey’s biweight. The second set of comparisons is between S power divergence estimation and numerical minimization. Monitoring these two procedures in terms of breakdown point shows that the numerical minimization yields a procedure with larger robust residuals and a lower empirical breakdown point, thus providing an estimate of <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> leading to more efficient parameter estimates.https://www.mdpi.com/1099-4300/22/4/399estimation of αmonitoringnumerical minimizationS-estimationTukey’s biweight |
| spellingShingle | Marco Riani Anthony C. Atkinson Aldo Corbellini Domenico Perrotta Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis Entropy estimation of α monitoring numerical minimization S-estimation Tukey’s biweight |
| title | Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis |
| title_full | Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis |
| title_fullStr | Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis |
| title_full_unstemmed | Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis |
| title_short | Robust Regression with Density Power Divergence: Theory, Comparisons, and Data Analysis |
| title_sort | robust regression with density power divergence theory comparisons and data analysis |
| topic | estimation of α monitoring numerical minimization S-estimation Tukey’s biweight |
| url | https://www.mdpi.com/1099-4300/22/4/399 |
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