Review Articles

The balance property in neural network modelling

Mario V. Wüthrich

RiskLab, Department of Mathematics, ETH Zurich, Zurich, Switzerland

Pages 1-9 | Received 09 Oct. 2019, Accepted 15 Jan. 2021, Published online: 21 Feb. 2021,
  • Abstract
  • Full Article
  • References
  • Citations

In estimation and prediction theory, considerable attention is paid to the question of having unbiased estimators on a global population level. Recent developments in neural network modelling have mainly focused on accuracy on a granular sample level, and the question of unbiasedness on the population level has almost completely been neglected by that community.We discuss this question within neural network regression models, and we provide methods of receiving unbiased estimators for these models on the global population level.


  • Breiman, L. (2001). Statistical modeling: The two cultures. Statistical Science, 16(3), 199231. [Crossref], [Web of Science ®][Google Scholar]
  • Breiman, L., Friedman, J. H., Olshen, R. A., & Stone, C. J. (1984). Classification and regression trees. Wadsworth Statistics/Probability Series[Google Scholar]
  • Bühlmann, H., & Gisler, A. (2005). A course in credibility theory and its applications. Springer[Google Scholar]
  • CASdatasets Package Vignette (2018). Version 1.0-8, May 20, 2018. [Google Scholar]
  • Charpentier, A. (2015). Computational actuarial science with R. CRC Press[Google Scholar]
  • Cheng, X., Jin, Z., & Yang, H. (2020). Optimal insurance strategies: A hybrid deep learning Markov chain approximation approach. ASTIN Bulletin, 50(2), 449477. [Crossref], [Web of Science ®][Google Scholar]
  • Cox, D. R. (1958). The regression analysis of binary sequences. Journal of the Royal Statistical Society: Series B (Methodological), 20(2), 215232. [Crossref], [Web of Science ®][Google Scholar]
  • Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2(4), 303314. [Crossref][Google Scholar]
  • Gabrielli, A. (2020). A neural network boosted double overdispersed Poisson claims reserving model. ASTIN Bulletin, 50(1), 2560. [Crossref], [Web of Science ®][Google Scholar]
  • Glorot, X., & Bengio, Y. (2010). Understanding the difficulty of training deep feedforward neural networks. Proceedings of Machine Learning Research, 9, 249256. Proceedings of the thirteenth international conference on artificial intelligence and statistics. [Google Scholar]
  • Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep learning. MIT Press[Google Scholar]
  • Green, P. J. (1984). Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives. Journal of the Royal Statistical Society: Series B (Methodological), 46(2), 149170. [Crossref], [Web of Science ®][Google Scholar]
  • Hornik, K., Stinchcombe, M., & White, H. (1989). Multilayer feedforward networks are universal approximators. Neural Networks, 2(5), 359366. [Crossref], [Web of Science ®][Google Scholar]
  • LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep learning. Nature, 521(7553), 436444. [Crossref], [Web of Science ®][Google Scholar]
  • Lee, G. Y., Manski, S., & Maiti, T. (2020). Actuarial applications of word embedding models. ASTIN Bulletin, 50(1), 124. [Crossref], [Web of Science ®][Google Scholar]
  • McCullagh, P., & Nelder, J. A. (1983). Generalized linear models. Chapman & Hall[Crossref][Google Scholar]
  • Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society. Series A (General), 135(3), 370384. [Crossref], [Web of Science ®][Google Scholar]
  • Noll, A., Salzmann, R., & Wüthrich, M. V. (2018). Case study: French motor third-party liability claims. SSRN Manuscript ID 3164764. Version March 4, 2020. [Crossref][Google Scholar]
  • Quinlan, J. R. (1992). Learning with continuous classes. In Proceedings of the 5th Australian joint conference on artificial intelligence (pp. 343–348). Singapore: World Scientific. [Google Scholar]
  • Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323(6088), 533536. [Crossref], [Web of Science ®][Google Scholar]
  • Schmidhuber, J. (2015). Deep learning in neural networks: An overview. Neural Networks, 61, 85117. [Crossref], [Web of Science ®][Google Scholar]
  • Shmueli, G. (2010). To explain or to predict? Statistical Science, 25(3), 289310. [Crossref], [Web of Science ®][Google Scholar]
  • Wang, C., Venkatesh, S. S., & Judd, J. S. (1994). Optimal stopping and effective machine complexity in learning. In Advances in neural information processing systems (NIPS'6) (pp. 303–310). [Google Scholar]
  • Wang, Y., & Witten, I. H. (1997). Inducing model trees for continuous classes. In Proceedings of the ninth European conference on machine learning (pp. 128–137). [Google Scholar]
  • Wüthrich, M. V., & Buser, C. (2016). Data analytics for non-life insurance pricing. SSRN Manuscript ID 2870308, Version of September 10, 2020. [Crossref][Google Scholar]
  • Wüthrich, M. V., & Merz, M. (2019). Editorial: Yes we CANN! ASTIN Bulletin, 49(1), 13. [Crossref], [Web of Science ®][Google Scholar]

To cite this article: Mario V. Wüthrich (2021): The balance property in neural network modelling,Statistical Theory and Related Fields, DOI: 10.1080/24754269.2021.1877960
To link to this article: