A Tweedie Compound Poisson Model in Reproducing Kernel Hilbert Space

References

• Allen, G. I. (2013), “Automatic Feature Selection via Weighted Kernels and Regularization,” Journal of Computational and Graphical Statistics, 22, 284–299. DOI: 10.1080/10618600.2012.681213.
   Web of Science ®Google Scholar
• Argyriou, A., Hauser, R., Micchelli, C. A., and Pontil, M. (2006), “A DC-Programming Algorithm for Kernel Selection,” in Proceedings of the 23rd International Conference on Machine Learning, pp. 41–48. DOI: 10.1145/1143844.1143850.
   Google Scholar
• Armijo, L. (1966), “Minimization of Functions Having Lipschitz Continuous First Partial Derivatives,” Pacific Journal of Mathematics, 16, 1–3. DOI: 10.2140/pjm.1966.16.1.
   Web of Science ®Google Scholar
• Blier-Wong, C., Cossette, H., Lamontagne, L., and Marceau, E. (2021), “Machine Learning in P&C Insurance: A Review for Pricing and Reserving,” Risks, 9, 4.
   Web of Science ®Google Scholar
• Brent, R. P. (2013) Algorithms for Minimization Without Derivatives, Chelmsford, MA: Courier Corporation.
   Google Scholar
• Broyden, C. G. (1970), “The Convergence of a Class of Double-Rank Minimization Algorithms 1. General Considerations,” IMA Journal of Applied Mathematics, 6, 76–90. DOI: 10.1093/imamat/6.1.76.
   Google Scholar
• Cao, B., Shen, D., Sun, J.-T., Yang, Q., and Chen, Z. (2007), “Feature Selection in a Kernel Space,” in Proceedings of the 24th International Conference on Machine Learning, pp. 121–128. DOI: 10.1145/1273496.1273512.
   Google Scholar
• Chen, J., Zhang, C., Kosorok, M. R., and Liu, Y. (2018), “Double Sparsity Kernel Learning with Automatic Variable Selection and Data Extraction,” Statistics and its Interface, 11, 401–420. DOI: 10.4310/SII.2018.v11.n3.a1.
   PubMed Web of Science ®Google Scholar
• Dons, K., Bhattarai, S., Meilby, H., Smith-Hall, C., and Panduro, T. E. (2016), “Indirect Approach for Estimation of Forest Degradation in Non-Intact Dry Forest: Modelling Biomass Loss with Tweedie Distributions,” Carbon Balance and Management, 11, 1–10. DOI: 10.1186/s13021-016-0051-z.
   PubMed Web of Science ®Google Scholar
• Dunn, P. K. (2004), “Occurrence and Quantity of Precipitation Can be Modelled Simultaneously,” International Journal of Climatology: A Journal of the Royal Meteorological Society, 24, 1231–1239. DOI: 10.1002/joc.1063.
   Web of Science ®Google Scholar
• Dunn, P. K., and Smyth, G. K. (2005), “Series Evaluation of Tweedie Exponential Dispersion Model Densities,” Statistics and Computing, 15, 267–280. DOI: 10.1007/s11222-005-4070-y.
   Web of Science ®Google Scholar
• Duvenaud, D. (2014), “The Kernel Cookbook: Advice on Covariance Functions,” Available at https://www.cs.toronto.edu/%7Eduvenaud/cookbook/.
   Google Scholar
• Dzupire, N. C., Ngare, P., and Odongo, L. (2018), “A Poisson-Gamma Model for Zero Inflated Rainfall Data,” Journal of Probability and Statistics, 2018, 1–12. DOI: 10.1155/2018/1012647.
   Web of Science ®Google Scholar
• El-Shaarawi, A. H., Zhu, R., and Joe, H. (2011), “Modelling Species Abundance using the Poisson–Tweedie Family,” Environmetrics, 22, 152–164. DOI: 10.1002/env.1036.
   Web of Science ®Google Scholar
• Fan, J., and Fan, Y. (2008), “High Dimensional Classification using Features Annealed Independence Rules,” Annals of statistics, 36, 2605–2637.
   PubMed Web of Science ®Google Scholar
• Fletcher, R. (1970),“A New Approach to Variable Metric Algorithms,” The Computer Journal, 13, 317–322. DOI: 10.1093/comjnl/13.3.317.
   Web of Science ®Google Scholar
• Fontaine, S., Yang, Y., Qian, W., Gu, Y., and Fan, B. (2020), “A Unified Approach to Sparse Tweedie Modeling of Multisource Insurance Claim Data,” Technometrics, 62, 339–356. DOI: 10.1080/00401706.2019.1647881.
   Web of Science ®Google Scholar
• Foster, S. D., and Bravington, M. V. (2013), “A Poisson–Gamma Model for Analysis of Ecological Non-negative Continuous Data,” Environmental and Ecological Statistics, 20, 533–552. DOI: 10.1007/s10651-012-0233-0.
   Web of Science ®Google Scholar
• Frees, E. W., Meyers, G., and Cummings, A. D. (2011), “Summarizing Insurance Scores using a Gini Index,” Journal of the American Statistical Association, 106, 1085–1098. DOI: 10.1198/jasa.2011.tm10506.
   Web of Science ®Google Scholar
• Friedman, J. H. (2001), “Greedy Function Approximation: A Gradient Boosting Machine,” Annals of Statistics, 29, 1189–1232.
   Web of Science ®Google Scholar
• Gilad-Bachrach, R., Navot, A., and Tishby, N. (2004), “Margin based Feature Selection – Theory and Algorithms,” in Proceedings of the Twenty-First International Conference on Machine Learning, 43. DOI: 10.1145/1015330.1015352.
   Google Scholar
• Giner, G., and Smyth, G. K. (2016), “statmod: Probability Calculations for the Inverse Gaussian Distribution,” R Journal, 8, 339–351. DOI: 10.32614/RJ-2016-024.
   Web of Science ®Google Scholar
• Goldfarb, D. (1970), “A Family of Variable-Metric Methods Derived by Variational Means,” Mathematics of Computation, 24, 23–26. DOI: 10.1090/S0025-5718-1970-0258249-6.
   Web of Science ®Google Scholar
• Grandvalet, Y., and Canu, S. (2002), “Adaptive Scaling for Feature Selection in svms,” in Advances in Neural Information Processing Systems (Vol. 15).
   Google Scholar
• Halder, A., Mohammed, S., Chen, K., and Dey, D. (2019), “Spatial Risk Estimation in Tweedie Compound Poisson Double Generalized Linear Models,” arXiv preprint arXiv:1912.12356.
   Google Scholar
• Hasan, M. M., and Dunn, P. K. (2011), “Two Tweedie Distributions that are Near-Optimal for Modelling Monthly Rainfall in Australia,” International Journal of Climatology, 31, 1389–1397. DOI: 10.1002/joc.2162.
   Web of Science ®Google Scholar
• Hastie, T., Tibshirani, R., Friedman, J. H., and Friedman, J. H. (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Vol. 2), New York: Springer.
   Google Scholar
• Hastie, T. J., and Tibshirani, R. J. (1990), Generalized Additive Models (Vol. 43), Boca Raton, FL: CRC Press.
   Google Scholar
• Islam, A. R. M. T., Hasanuzzaman, M., Shammi, M., Salam, R., Bodrud-Doza, M., Rahman, M. M., Mannan, M. A., and Huq, S. (2021), “Are Meteorological Factors Enhancing COVID-19 Transmission in Bangladesh? Novel findings from a Compound Poisson Generalized Linear Modeling Approach,” Environmental Science and Pollution Research, 28, 11245–11258. DOI: 10.1007/s11356-020-11273-2.
   PubMed Web of Science ®Google Scholar
• Jørgensen, B. (1987), “Exponential Dispersion Models,” Journal of the Royal Statistical Society, Series B, 49, 127–162. DOI: 10.1111/j.2517-6161.1987.tb01685.x.
   Google Scholar
• Jørgensen, B. (1997), The Theory of Dispersion Models, Boca Raton, FL: CRC Press.
   Google Scholar
• Jørgensen, B., and de Souza, M. C. (1994), “Fitting Tweedie’s Compound Poisson Model to Insurance Claims Data,” Scandinavian Actuarial Journal, 1994, 69–93. DOI: 10.1080/03461238.1994.10413930.
   Google Scholar
• Kurz, C. F. (2017), “Tweedie Distributions for Fitting Semicontinuous Health Care Utilization Cost Data,” BMC Medical Research Methodology, 17, 1–8. DOI: 10.1186/s12874-017-0445-y.
   PubMed Web of Science ®Google Scholar
• Lee, S. C., and Lin, S. (2018), “Delta Boosting Machine with Application to General Insurance,” North American Actuarial Journal, 22, 405–425. DOI: 10.1080/10920277.2018.1431131.
   Web of Science ®Google Scholar
• Li, F., Yang, Y., and Xing, E. (2005), “From Lasso Regression to Feature Vector Machine,” Advances in Neural Information Processing Systems, 18, 779–786.
   Google Scholar
• Lin, Y., and Zhang, H. H. (2006), “Component Selection and Smoothing in Multivariate Nonparametric Regression,” The Annals of Statistics, 34, 2272–2297. DOI: 10.1214/009053606000000722.
   Web of Science ®Google Scholar
• Mack, T. (1993), “Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates,” ASTIN Bulletin: The Journal of the IAA, 23, 213–225. DOI: 10.2143/AST.23.2.2005092.
   Google Scholar
• Meyers, G. G., and Shi, P. (2011), “Loss Reserving Data Pulled from NAIC Schedule P.” Available at https://www.casact.org/publications-research/research/research-resources/loss-reserving-data-pulled-naic-schedule-p.
   Google Scholar
• Moshitch, D., and Nelken, I. (2014), “Using Tweedie Distributions for Fitting Spike Count Data,” Journal of Neuroscience Methods, 225, 13–28. DOI: 10.1016/j.jneumeth.2014.01.004.
   PubMed Web of Science ®Google Scholar
• NAIC. (2021), “Data Products – Schedule P,” Available at https://content.naic.org/prod_serv_idp_sched_p.htm.
   Google Scholar
• Nocedal, J., and Wright, S. (2006), Numerical Optimization, New York: Springer.
   Google Scholar
• Ohlsson, E., and Johansson, B. (2010), Non-Life Insurance Pricing with Generalized Linear Models (Vol. 2), Berlin: Springer.
   Google Scholar
• Peters, G. W., Shevchenko, P. V., and Wüthrich, M. V. (2008), “Model Risk in Claims Reserving within Tweedie’s Compound Poisson Models,” ASTIN Bulletin (to appear). DOI: 10.2143/AST.39.1.2038054.
   Google Scholar
• Peters, G. W., Shevchenko, P. V., and Wüthrich, M. V. (2009), “Model Uncertainty in Claims Reserving within Tweedie’s Compound Poisson Models,” ASTIN Bulletin: The Journal of the IAA, 39, 1–33.
   Web of Science ®Google Scholar
• Qian, W., Yang, Y., and Zou, H. (2016), “Tweedie’s Compound Poisson Model with Grouped Elastic Net,” Journal of Computational and Graphical Statistics, 25, 606–625. DOI: 10.1080/10618600.2015.1005213.
   Web of Science ®Google Scholar
• Rahimi, A., and Recht. (2007), “Random Features for Large-Scale Kernel Machines,” in Advances in Neural Information Processing Systems (Vol. 3), p. 5.
   Google Scholar
• Rasmussen, C. E. (2003), “Gaussian Processes in Machine Learning,” in Summer School on Machine Learning, eds. O. Bousquet, U. Luxburg, and G. Rätsch, pp. 63–71, Berlin: Springer.
   Google Scholar
• Rudi, A., Camoriano, R., and Rosasco, L. (2015), “Less is More: Nyström Computational Regularization,” in Advances in Neural Information Processing Systems, pp. 1657–1665.
   Google Scholar
• Shanno, D. F. (1970), “Conditioning of Quasi-Newton Methods for Function Minimization,” Mathematics of Computation, 24, 647–656. DOI: 10.1090/S0025-5718-1970-0274029-X.
   Web of Science ®Google Scholar
• Shi, P. (2014), “A Copula Regression for Modeling Multivariate Loss Triangles and Quantifying Reserving Variability,” ASTIN Bulletin: The Journal of the IAA, 44, 85–102. DOI: 10.1017/asb.2013.23.
   Web of Science ®Google Scholar
• Shi, P. (2016), “Insurance Ratemaking Using a Copula-based Multivariate Tweedie Model,” Scandinavian Actuarial Journal, 2016, 198–215.
   Web of Science ®Google Scholar
• Shi, P., Feng, X., and Boucher, J.-P. (2016), “Multilevel Modeling of Insurance Claims Using Copulas,” The Annals of Applied Statistics, 10, 834–863. DOI: 10.1214/16-AOAS914.
   Web of Science ®Google Scholar
• Shono, H. (2008), “Application of the Tweedie Distribution to Zero-Catch Data in CPUE Analysis,” Fisheries Research, 93, 154–162. DOI: 10.1016/j.fishres.2008.03.006.
   Web of Science ®Google Scholar
• Smyth, G., Hu, Y., Dunn, P., Phipson, B. and shun Chen, Y. (2021), Statistical Modeling. R package version 1.4.36. Available at https://cran.r-project.org/package=statmod.
   Google Scholar
• Smyth, G., and Jorgensen, B. (2002), “Fitting Tweedie’s Compound Poisson Model to Insurance Claims Data: Dispersion Modelling,” ASTIN Bulletin, 32, 143–157. DOI: 10.2143/AST.32.1.1020.
   Google Scholar
• Smyth, G. K. (1996), “Regression Analysis of Quantity Data with Exact Zeros,” in Proceedings of the second Australia–Japan workshop on Stochastic Models in Engineering, Technology and Management, pp. 572–580.
   Google Scholar
• Sriram, K., and Shi, P. (2020), “Stochastic Loss Reserving: A New Perspective from a Dirichlet Model,” Journal of Risk and Insurance, 88, 195–230. DOI: 10.1111/jori.12311.
   Web of Science ®Google Scholar
• Taylor, G. (2019), “Loss Reserving Models: Granular and Machine Learning Forms,” Risks, 7, 82. DOI: 10.3390/risks7030082.
   Web of Science ®Google Scholar
• Taylor, G., and McGuire, G. (2016), “Stochastic Loss Reserving Using Generalized Linear Models,” CAS Monograph, 3, 1–112.
   Google Scholar
• Tweedie, M. (1984), “An Index which Distinguishes between Some Important Exponential Families,” in Statistics: Applications and New Directions: Proc. Indian Statistical Institute Golden Jubilee International Conference, pp. 579–604.
   Google Scholar
• Vapnik, V. (2013), The Nature of Statistical Learning Theory, New York: Springer.
   Google Scholar
• Wahba, G. (1990), Spline Models for Observational Data (Vol. 59), Philadelphia, PA: SIAM.
   Google Scholar
• Weston, J., Mukherjee, S., Chapelle, O., Pontil, M., Poggio, T., and Vapnik, V. (2000), “Feature Selection for SVMs,” in Advances in Neural Information Processing Systems (Vol. 13), eds. T. Leen, T. Dietterich and V. Tresp, MIT Press.
   Google Scholar
• Wolfe, P. (1971), “Convergence Conditions for Ascent Methods. II: Some Corrections,” SIAM Review, 13, 185–188. DOI: 10.1137/1013035.
   Web of Science ®Google Scholar
• Wood, S. (2021), Mixed GAM Computation Vehicle with Automatic Smoothness Estimation. R package version 1.8-36. https://cran.r-project.org/package=mgcv.
   Google Scholar
• Wood, S. N. (2011), “Fast Stable Restricted Maximum Likelihood and Marginal Likelihood Estimation of Semiparametric Generalized Linear Models,” Journal of the Royal Statistical Society, Series B, 73, 3–36. DOI: 10.1111/j.1467-9868.2010.00749.x.
   Google Scholar
• Wüthrich, M. V. (2003), “Claims Reserving using Tweedie’s Compound Poisson Model,” ASTIN Bulletin: The Journal of the IAA, 33, 331–346. DOI: 10.1017/S0515036100013490.
   Google Scholar
• Yang, L., Lv, S., and Wang, J. (2016), “Model-Free Variable Selection in Reproducing Kernel Hilbert Space,” The Journal of Machine Learning Research, 17, 2885–2908.
   Web of Science ®Google Scholar
• Yang, Y., Luo, R., and Liu, Y. (2019), “Adversarial Variational Bayes Methods for Tweedie Compound Poisson Mixed Models,” in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3377–3381. IEEE.
   Google Scholar
• Yang, Y., Qian, W., and Zou, H. (2016), TDboost: A Boosted Tweedie Compound Poisson Model. R package version 1.2. Available at https://CRAN.R-project.org/package=TDboost.
   Google Scholar
• Yang, Y., Qian, W., and Zou, H. (2018), “Insurance Premium Prediction via Gradient Tree-Boosted Tweedie Compound Poisson Models,” Journal of Business & Economic Statistics, 36, 456–470.
   Web of Science ®Google Scholar
• Ye, C., Zhang, L., Han, M., Yu, Y., Zhao, B., and Yang, Y. (2018), “Combining Predictions of Auto Insurance Claims,” arXiv preprint arXiv:1808.08982.
   Google Scholar
• Yip, K. C., and Yau, K. K. (2005), “On Modeling Claim Frequency Data in General Insurance with Extra Zeros,” Insurance: Mathematics and Economics, 36, 153–163. DOI: 10.1016/j.insmatheco.2004.11.002.
   Web of Science ®Google Scholar
• Zhang, Y. (2013), “Likelihood-based and Bayesian Methods for Tweedie Compound Poisson Linear Mixed Models,” Statistics and Computing, 23, 743–757. DOI: 10.1007/s11222-012-9343-7.
   Web of Science ®Google Scholar
• Zhou, H., Qian, W., and Yang, Y. (2020), “Tweedie Gradient Boosting for Extremely Unbalanced Zero-Inflated Data,” Communications in Statistics-Simulation and Computation, 59, 5507–5529.
   Google Scholar