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Valid model-free prediction of future insurance claims. (English) Zbl 07469929

The authors consider conformal prediction as a model-free method to provide provably valid predictions for the size of future insurance claims uniformly over all sample sizes and all distributions.
Let \(X_{n}\) \((n \in \mathbb{N})\) be real-valued random variables such that the joint distribution of any finite subsequence is permutation invariant. For \(M\colon [\mathbb{R}]^{<\infty} \times \mathbb{R} \rightarrow \mathbb{R}\), called nonconformity measure, given a set of observations \(\mathcal{X}_{n} = \{x_{0},x_{1},\dots, x_{n-1}\}\), the plausibility function \(\textrm{pl}_{\mathcal{X}_{n}}\colon \mathbb{R} \rightarrow \mathbb{R}\) is calculated as follows:
1.
Let \(x \in \mathbb{R}\) and set \(x_{n} = x\), \(\mathcal{X}_{n+1} = \mathcal{X}_{n}\cup \{x\}\).
2.
For every \(i\leq n\), set \(m_{i} = M(\mathcal{X}_{n+1}\setminus \{x_{i}\},x_{i})\).
3.
Set \(\textrm{pl}_{\mathcal{X}_{n}}(x) = \frac{1}{n+1}\sum_{i\leq n} \mathbb{I}_{[m_{i}\geq m_{n}]}\).
The authors note that, in non-degenerate cases, \(\textrm{pl}_{\mathcal{X}_{n}}(X_{n+1})\) is uniformly distributed on \(\{1/(n+1),2/(n+1),\dots,1\}\). Hence, for every \(\alpha \in (0,1)\), \[C_{\alpha}(\mathcal{X}_{n}) = \{x\colon \textrm{pl}_{\mathcal{X}_{n}}(x) > \lfloor(n+1)\alpha\rfloor/(n+1) \}\] satisfies that the probability of \(X_{n+1}\notin C_{\alpha}(\mathcal{X}_{n}) \) does not exceed \(\alpha\), which makes \(C_{\alpha}(\mathcal{X}_{n})\) the \(100(1-\alpha)\%\) conformal prediction region.
The authors calculate \(m_{i}\) explicitly in the case \(M\) is the sample cumulative density function of the form \(M(\mathcal{X}_{n},x) = \frac{1}{n}\sum_{i<n}K(x,x_{i},h)\) where \(K(\cdot,\theta,h)\) is a kernel distribution function with parameter \(\theta \in \mathbb{R}\) and bandwidth \(h\), and observe that if \(x \mapsto K(x,x,h)\) is constant then \( \textrm{pl}_{\mathcal{X}_{n}}(x) = \frac{1}{n}\sum_{i<n}\mathbb{I}_{[x_{i}\geq x]}\) and \(C_{\alpha}(\mathcal{X}_{n}) = [0,x_{(k)})\) where \(x_{(k)}\) is the \(k^{\textrm{th}}\) largest element of \(\mathcal{X}_{n}\) with \(k = \min\{n,\lfloor (n+1)(1-\alpha)\rfloor+1\}\).
The authors use fire claims data and car injury claims data to calculate conformal prediction intervals and compare them to prediction intervals based on other non-parametric methods. Possible extensions of the method to conditional predictions are also considered.

MSC:

91G05 Actuarial mathematics
62G30 Order statistics; empirical distribution functions
60G25 Prediction theory (aspects of stochastic processes)
62M20 Inference from stochastic processes and prediction
62G05 Nonparametric estimation
62G07 Density estimation

Software:

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References:

[1] Brazauskas, Y.,; Kleefeld., A., Modeling severity and measuring tail risk of Norwegian fire claims, North American Actuarial Journal, 20, 1, 1-16 (2016) · Zbl 1414.62415
[2] Cella, L.; Martin, R. (2019)
[3] de Jong, P.; Heller, P. Z., Generalized linear models for insurance data (2008), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1142.91046
[4] Fligner, M.; Wolfe., D. A., Some applications of sample analogues to the probability integral transformation and a coverage property, The American Statistician, 30, 2, 78-85 (1976) · Zbl 0369.62015
[5] Foygel-Barber, R.; Candes, E. J.; Ramadas, A.; Tibshirani., R. J., Conformal prediction under covariate shift, arXiv:1904.06019 (2019)
[6] Foygel-Barber, R.; Candes, E. J.; Ramadas, A.; Tibshirani., R. J., The limits of distribution-free conditional predictive inference (2019)
[7] Frees, E. W.; Derrig, R. A.; Meyers., G., Predictive modeling applications in actuarial science: Vol. I. Predictive modeling techniques (2014), Cambridge: Cambridge University Press, Cambridge
[8] Frey, J., Data-driven nonparametric prediction intervals, Journal of Statistical Planning and Inference, 143, 1039-48 (2013) · Zbl 1428.62183
[9] Gan, G.; Valdez., E. A., Data clustering with actuarial applications, North American Actuarial Journal (2019) · Zbl 1454.91186
[10] Ghahari, A.; Newlands, N. K.; Lyubchich, V.; Gel., Y. R., Deep learning at the interface of agricultural insurance risk and spatio-temporal uncertainty in weather extremes, North American Actuarial Journal, 23, 4, 535-50 (2019) · Zbl 1429.91278
[11] Guan, L., Conformal prediction with localization, arXiv:1908.08558 (2019)
[12] Hong, L.; Kuffner, T.; Martin., R., On prediction of future insurance claims when the model is uncertain, Variance, 12, 1, 90-99 (2018)
[13] Hong, L.; Martin., R., A flexible Bayesian nonparametrics model for predicting future insurance claims, North American Actuarial Journal, 21, 228-41 (2017) · Zbl 1414.91201
[14] Hong, L.; Martin., R., Dirichlet process mixture models for insurance loss data, Scandinavian Actuarial Journal, 6, 545-54 (2018) · Zbl 1416.91188
[15] Hong, L.; Martin., R., Real-time Bayesian nonparametric prediction of solvency risk, Annals of Actuarial Science, 13, 67-79 (2019)
[16] Hong, L.; Martin, R., Model misspecification, Bayesian versus credibility estimation, and Gibbs posteriors, Scandinavian Actuarial Journal (2020) · Zbl 1448.91261
[17] Jeon, Y.; Kim., J. H. T., A gamma kernel density estimation for insurance loss data, Insurance: Mathematics and Economics, 53, 569-79 (2013) · Zbl 1290.62099
[18] Kallenberg, O., Foundations of modern probability (2002), New York: Springer, New York · Zbl 0996.60001
[19] Kleijn, B. J. K.; Van der Vaart., A. W., The Bernstein-Von-Mises theorem under misspecification, Electronic Journal of Statistics, 6, 354-81 (2012) · Zbl 1274.62203
[20] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss models: From data to decisions (2008), Hoboken, NJ: Wiley, Hoboken, NJ · Zbl 1159.62070
[21] Lee, S. C. K.; Lin., X. S., Delta boosting machine with application to general insurance, North American Actuarial Journal, 22, 3, 405-25 (2018) · Zbl 1416.91199
[22] Lei, J.; Wasserman., L., Distribution-free prediction bands for nonparametric regression, Journal of Royal Statistical Society-Series B, 76, 71-96 (2014) · Zbl 1411.62103
[23] Liu, K.; Tan, K. S., Real-time valuation of large variable annuity portfolios: A Green Mesh Approach, North American Actuarial Journal (2020)
[24] Lockhart, R.; Taylor, J.; Tibshirani, R. J.; Tibshirani., R., A significance test for the lasso, Annals of Statistics, 42, 2, 413-68 (2014) · Zbl 1305.62254
[25] Martin, R., A statistical inference course based on p-values, The American Statistician, 71, 128-36 (2017)
[26] Martin, R., False confidence, non-additive beliefs, and valid statistical inference, International Journal of Approximate Reasoning, 113, 39-73 (2019) · Zbl 1471.62236
[27] Martin, R.; Lingham., R. T., Prior-free probabilistic prediction of future observations, Technometrics, 58, 2, 226-35 (2016)
[28] Martin, R.; Liu., C., Inferential models: A framework for prior-free posterior probabilistic inference, Journal of the American Statistical Association, 108, 501, 301-13 (2013) · Zbl 06158344
[29] Martin, R.; Liu., C., A note on p-values interpreted as plausibilities, Statistica Sinica, 24, 1703-16 (2014) · Zbl 1480.62010
[30] Martin, R.; Liu, C., Inferential models: Reasoning with uncertainty (2015), Boca Raton, FL: Chapman & Hall/CRC Press
[31] Mdziniso, N. C.; Cooray., K., Odd Pareto families of distributions for modeling loss payment data, Scandinavian Actuarial Journal, 1, 42-63 (2018) · Zbl 1416.91208
[32] Norwegian fire claims data. 1990. Accessed October 12, 2019.
[33] Rempala, G. A.; Derrig., R. A., Modeling hidden exposures in claim severity via the EM algorithm, North American Actuarial Journal, 9, 2, 108-28 (2005) · Zbl 1085.62515
[34] Schervish, M. J., Theory of statistics (1995), New York: Springer, New York · Zbl 0834.62002
[35] Shafer, G.; Vovk, V., A tutorial on conformal prediction, Journal of Machine Learning, 9, 371-421 (2008) · Zbl 1225.68215
[36] Sheather, S., Density estimation, Statistical Science, 19, 588-597 (2004) · Zbl 1100.62558
[37] Solvency II (2009)
[38] Syring, N.; Hong, L.; Martin., R., Gibbs posterior inference on value-at-risk, Scandinavian Actuarial Journal, 2019, 7, 548-57 (2019) · Zbl 1422.91376
[39] Vovk, V., Conditional validity of inductive conformal predictors, Machine Learning, 92, 475-90 (2013) · Zbl 1273.68307
[40] Vovk, V.; Gammerman, A.; Shafer, G., Algorithmic learning in a random world (2005), New York: Springer, New York · Zbl 1105.68052
[41] Werner, G.; Modlin, C., Basic ratemaking (2010), Arlington, VA: Casualty Actuarial Society, Arlington, VA
[42] Wilks, W., Determination of sample sizes for setting tolerance limits, Annals of Mathematical Statistics, 12, 91-96 (1941) · JFM 67.0481.04
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