Abstract
In the paper, we introduce novel model selection measures based on Lorenz zonoids which, differently from measures based on correlations, are based on a mutual notion of variability and are more robust to the presence of outlying observations. By means of Lorenz zonoids, which in the univariate case correspond to the Gini coefficient, the contribution of each explanatory variable to the predictive power of a linear model can be measured more accurately. Exploiting Lorenz zonoids, we develop a Marginal Gini Contribution measure that allows to measure the absolute explanatory power of any covariate, and a Partial Gini Contribution measure that allows to measure the additional contribution of a new covariate to an existing model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods & Research, 33(2), 261–304.
Calabrese, R., & Giudici, P. (2015). Estimating bank default with generalised extreme value models. Journal of the Operational Research Society, 66, 1783–1792.
Dall’Aglio, M., & Scarsini, M. (2003). Zonoids, linear dependence, and size-biased distributions on the simplex. Advances in Applied Probability, 35, 871–884.
Diebold, F. X., & Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business and Economic Statistics, 13(3), 253–263.
Fantazzini, D., Dalla Valle, L., Giudici, P. (2008). Copulae and operational risk. International Journal of Risk Assessment and Management, 9, 238–257.
Figini, S., & Giudici, P. (2011). Statistical merging of rating models. Journal of the Operational Research Society, 62, 1067–1074.
Giudici, P. (2003). Applied data mining: statistical methods for business and industry, 1st edn. Hoboken: Wiley.
Giudici, P., & Bilotta, A. (2004). Modelling operational losses: a Bayesian approach. Quality and Reliability Engineering International, 20, 407–417.
Giudici, P., & Raffinetti, E. (2011). On the Gini measure decomposition. Statistics & Probability Letters, 81(1), 133–139.
Giudici, P., & Abu-Hashish, I. (2019). What determines bitcoin exchange prices? A network VAR approach. Finance Research Letters, 28, 309–318.
Hand, D., Mannilla, H., Smyth, P. (2001). Principles of data mining adaptive computation and machine learning series, 1st edn. Boston: MIT Press.
Koshevoy, G. (1995). Multivariate Lorenz majorization. Social Choice and Welfare, 12(1), 93–102.
Koshevoy, G., & Mosler, K. (1996). The Lorenz zonoids of a multivariate distribution. Journal of the American Statistical Association, 91(434), 873–882.
Koshevoy, G., & Mosler, K. (2007). Multivariate Lorenz dominance based on zonoids. AStA Advances in Statistical Analysis, 91(1), 57–76.
Lerman, R., & Yitzhaki, S. (1984). A note on the calculation and interpretation of the Gini index. Economics Letters, 15(3-4), 363–368.
Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Journal of the American Statistical Association, 9(70), 209–219.
Mosler, K. (1994). Majorization in economic disparity measures. Linear Algebra and its applications, 119(1), 91–114.
Muliere, P., & Petrone, S. (1992). Generalized Lorenz curve and monotone dependence orderings. Metron, L(3-4), 19–38.
Raffinetti, E., & Giudici, P. (2013). Lorenz zonoids and dependence measures: a proposal. In Torelli, N., Pesarin, F., Bar-Hen, A. (Eds.) Theoretical and applied statistics series: Studies in theoretical and applied statistics. Berlin: Springer.
Raffinetti, E., Siletti, E., Vernizzi, A. (2015). On the Gini coefficient normalization when attributes with negative values are considered. Statistical Methods & Applications, 24(3), 507–521.
Rousseeuw, P. J., & Leroy, A. M. (1987). Robust regression and outlier detection, Inc. New York: Wiley.
Acknowledgments
We would like to thank the anonymous referees and the editor for their useful comments and suggestions on the paper. The research in the paper was funded by the European Union’s Horizon 2020 research and innovation program under grant agreement No 825215 (Topic: ICT-35-2018 Type of action: CSA). While the paper is the result of a close cooperation between the two Authors, ER wrote Sections 2, 3 and 4.1; PG wrote Sections 1, 4.2 and 5.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Giudici, P., Raffinetti, E. Lorenz Model Selection. J Classif 37, 754–768 (2020). https://doi.org/10.1007/s00357-019-09358-w
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00357-019-09358-w