# In-Sample Hazard Forecasting Based on Survival Models with Operational Time

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. General Model

#### 2.2. Model on the Run-Off Triangle with Right-Truncation

## 3. Estimation of Baseline Hazard and Operational Time

- Estimate the (unstructured) conditional hazard by ${\widehat{\alpha}}^{\left[0\right]}(t,x)$ through Equation (7).
- Set ${\widehat{\phi}}^{\left[0\right]}\equiv 1$ and $r=1$.
- Estimate ${\widehat{\alpha}}_{0}^{\left[r\right]}$ through Equation (10) using ${\widehat{\phi}}^{[r-1]}$.
- Estimate ${\widehat{\phi}}^{\left[r\right]}$ by minimizing the loss in (11) numerically for every x using ${\widehat{\alpha}}_{0}^{[r-1]}$.
- Repeat steps 3 and 4 for $r=2,3,4,\cdots $ until the convergence criterion$${\int}_{0}^{\mathcal{T}}{\left({\widehat{\phi}}^{\left[r\right]}(x)-{\widehat{\phi}}^{[r-1]}(x)\right)}^{2}\mathrm{d}x<{10}^{-5}$$
- Set the final conditional hazard estimator to$$\widehat{\alpha}(t,x)=\frac{1}{\widehat{\phi}(x)}{\widehat{\alpha}}_{0}\left(\mathcal{T}-\frac{\mathcal{T}-t}{\widehat{\phi}(x)}\right),$$$$\begin{array}{cc}\hfill \widehat{\phi}& ={\widehat{\phi}}^{\left[{r}^{*}\right]},\hfill \end{array}$$$$\begin{array}{cc}\hfill {\widehat{\alpha}}_{0}& ={\widehat{\alpha}}_{0}^{\left[{r}^{*}\right]}.\hfill \end{array}$$The final estimator in (non-reversed) “forward time” is set to$${\widehat{\alpha}}^{f}(t,x)=\widehat{\alpha}(\mathcal{T}-t,x).$$

#### 3.1. Pre-Step: Unstructured Conditional Hazard

#### 3.2. Estimation of Baseline Hazard Given Operational Time

#### 3.3. Estimation of Operational Time Given Baseline Hazard

## 4. Estimating Outstanding Claim Amounts

## 5. Bandwidth Selection

## 6. Application: Estimation of Outstanding Liabilities

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Alternative Local Constant Estimators

## References

- Aalen, Odd O. 1980. A model for nonparametric regression analysis of counting processes. In Mathematical Statistics and Probability Theory. Lecture Notes in Statistics. Edited by Witold Klonecki, Andrzej Kozek and Jan Rosiński. New York: Springer, vol. 2, pp. 1–25. [Google Scholar]
- Andersen, Per K., Ørnulf Borgan, Richard D. Gill, and Niels Keiding. 1993. Statistical Models Based on Counting Processes. New York: Springer. [Google Scholar]
- Antonio, Katrien, and Richard Plat. 2014. Micro-level stochastic loss reserving for general insurance. Scandinavian Actuarial Journal 2014: 649–69. [Google Scholar] [CrossRef]
- Austin, Matthew D., and Rebecca A. Betensky. 2014. Eliminating bias due to censoring in kendall’s tau estimators for quasi-independence of truncation and failure. Computational Statistics & Data Analysis 73: 16–26. [Google Scholar]
- Avanzi, Benjamin, Bernard Wong, and Xinda Yang. 2016. A micro-level claim count model with overdispersion and reporting delays. Insurance: Mathematics and Economics 71: 1–14. [Google Scholar] [CrossRef] [Green Version]
- Badescu, Andrei L., X. Sheldon Lin, and Dameng Tang. 2016. A marked Cox model for the number of IBNR claims: Theory. Insurance: Mathematics and Economics 69: 29–37. [Google Scholar] [CrossRef]
- Baudry, Maximilien, and Christian Y. Robert. 2019. A machine learning approach for individual claims reserving in insurance. Applied Stochastic Models in Business and Industry 35: 1127–55. [Google Scholar] [CrossRef]
- Berkson, Joseph. 1980. Minimum chi-square, not maximum likelihood! The Annals of Statistics 8: 457–87. [Google Scholar] [CrossRef]
- Bischofberger, Stephan M., Munir Hiabu, and Alex Isakson. 2019. Continuous chain-ladder with paid data. Scandinavian Actuarial Journal. [Google Scholar] [CrossRef]
- Buckley, Jonathan, and Ian James. 1979. Linear regression with censored data. Biometrika 66: 429–36. [Google Scholar] [CrossRef]
- Bühlmann, Hans. 1970. Mathematical Methods in Risk Theory. Berlin: Springer. [Google Scholar]
- Cho, Youngjoo, Chen Hu, and Debashis Ghosh. 2018. Covariate adjustment using propensity scores for dependent censoring problems in the accelerated failure time model. Statistics in Medicine 37: 390–404. [Google Scholar] [CrossRef]
- Cox, David R. 1972. Regression models and life tables. Journal of the Royal Statistical Society: Series B 34: 187–220. [Google Scholar]
- Cox, David R., and David Oakes. 1984. Analysis of Survival Data, 1st ed. Boca Raton: Chapman & Hall/CRC. [Google Scholar]
- Crevecoeur, Jonas, Katrien Antonio, and Roel Verbelen. 2019. Modeling the number of hidden events subject to observation delay. European Journal of Operational Research 277: 930–44. [Google Scholar] [CrossRef] [Green Version]
- England, Peter D., and Richard J. Verrall. 2002. Stochastic claims reserving in general insurance. British Actuarial Journal 8: 443–544. [Google Scholar] [CrossRef]
- Feller, William. 1971. An Introduction to Probability Theory and Its Applications. New York: John Wiley & Sons, vol. 2. [Google Scholar]
- Fulcher, Isabel R., Eric Tchetgen Tchetgen, and Paige L. Williams. 2017. Mediation analysis for censored survival data under an accelerated failure time model. Epidemiology 28: 660–66. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Gabrielli, Andrea, Ronald Richman, and Mario V. Wüthrich. 2019. Neural network embedding of the over-dispersed Poisson reserving model. Scandinavian Actuarial Journal. [Google Scholar] [CrossRef]
- Gámiz, María Luz, Lena Janys, María Dolores Martínez-Miranda, and Jens Perch Nielsen. 2013. Bandwidth selection in marker dependent kernel hazard estimation. Computational Statistics & Data Analysis 68: 155–69. [Google Scholar]
- Hastie, Trevor, Robert Tibshirani, and Jerome Friedman. 2008. The Elements of Statistical Learning: Data Mining, Inference and Prediction. New York: Springer. [Google Scholar]
- Hiabu, Munir. 2017. On the relationship between classical chain ladder and granular reserving. Scandinavian Actuarial Journal 2017: 708–29. [Google Scholar] [CrossRef] [Green Version]
- Hiabu, Munir, Enno Mammen, María Dolores Martínez-Miranda, and Jens Perch Nielsen. 2016. In-sample forecasting with local linear survival densities. Biometrika 103: 843–59. [Google Scholar] [CrossRef]
- Huang, Jinlong, Chunjuan Qiu, Xianyi Wu, and Xian Zhou. 2015. An individual loss reserving model with independent reporting and settlement. Insurance: Mathematics and Economics 64: 232–45. [Google Scholar] [CrossRef]
- Jewell, William S. 1989. Predicting IBNYR events and delays I. Continuous time. ASTIN Bulletin 19: 25–55. [Google Scholar] [CrossRef] [Green Version]
- Jewell, William S. 1990. Predicting IBNYR events and delays II. Discrete time. ASTIN Bulletin 20: 93–111. [Google Scholar] [CrossRef] [Green Version]
- Kalbfleisch, John D., and Ross L. Prentice. 2002. The Statistical Analysis of Failure Time Data, 2nd ed. Wiley Series in Probability and Statistics; Hoboken: John Wiley & Sons. [Google Scholar]
- Kremer, Erhard. 1982. IBNR-claims and the two-way model of ANOVA. Scandinavian Actuarial Journal 1982: 47–55. [Google Scholar] [CrossRef]
- Kuang, Di, Bent Nielsen, and Jens Perch Nielsen. 2009. Chain-ladder as maximum likelihood revisited. Annals of Actuarial Science 4: 105–21. [Google Scholar] [CrossRef] [Green Version]
- Kuo, Kevin. 2019. Deeptriangle: A deep learning approach to loss reserving. Risks 7: 97. [Google Scholar] [CrossRef] [Green Version]
- Larsen, Christian Roholte. 2007. An individual claims reserving model. ASTIN Bulletin 37: 113–32. [Google Scholar] [CrossRef] [Green Version]
- Lee, Young K., Enno Mammen, Jens Perch Nielsen, and Byeong U. Park. 2015. Asymptotics for in-sample density forecasting. The Annals of Statistics 43: 620–51. [Google Scholar] [CrossRef]
- Lee, Young K., Enno Mammen, Jens Perch Nielsen, and Byeong U. Park. 2017. Operational time and in-sample density forecasting. The Annals of Statistics 45: 1312–41. [Google Scholar] [CrossRef] [Green Version]
- Li, Jialiang, and Baisuo Jin. 2018. Multi-threshold accelerated failure time model. The Annals of Statistics 46: 2657–82. [Google Scholar] [CrossRef]
- Linton, Oliver B., Enno Mammen, Jens Perch Nielsen, and Ingrid Van Keilegom. 2011. Nonparametric regression with filtered data. Bernoulli 17: 60–87. [Google Scholar] [CrossRef]
- Louis, Thomas A. 1981. Nonparametric analysis of an accelerated failure time model. Biometrika 68: 381–90. [Google Scholar] [CrossRef]
- Mack, Thomas. 1993. Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 23: 213–25. [Google Scholar] [CrossRef] [Green Version]
- Mammen, Enno, María Dolores Martínez-Miranda, and Jens Perch Nielsen. 2015. In-sample forecasting applied to reserving and mesothelioma. Insurance: Mathematics and Economics 61: 76–86. [Google Scholar] [CrossRef] [Green Version]
- Martin, Emily C., and Rebecca A. Betensky. 2005. Testing quasi-independence of failure and truncation times via conditional Kendall’s tau. Journal of the American Statistical Association 100: 484–92. [Google Scholar] [CrossRef]
- Martínez-Miranda, María Dolores, Jens Perch Nielsen, Stefan Sperlich, and Richard J. Verrall. 2013. Continuous chain ladder: Reformulating and generalising a classical insurance problem. Expert Systems with Applications 40: 5588–603. [Google Scholar] [CrossRef]
- Miller, Rupert G. 1976. Least squares regression with censored data. Biometrika 63: 449–64. [Google Scholar] [CrossRef]
- Nielsen, Jens Perch. 1998. Marker dependent kernel hazard estimation from local linear estimation. Scandinavian Actuarial Journal 1998: 113–24. [Google Scholar] [CrossRef]
- Nielsen, Jens Perch, and Oliver B. Linton. 1995. Kernel estimation in a non-parametric marker dependent hazard model. The Annals of Statistics 23: 1735–48. [Google Scholar] [CrossRef]
- Nielsen, Jens Perch, and Carsten Tanggaard. 2001. Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics 28: 675–98. [Google Scholar] [CrossRef]
- Norberg, Ragnar. 1993. Prediction of outstanding liabilities in non-life insurance. ASTIN Bulletin 23: 95–115. [Google Scholar] [CrossRef] [Green Version]
- Norberg, Ragnar. 1999. Prediction of outstanding liabilities II. Model variations and extensions. ASTIN Bulletin 29: 5–25. [Google Scholar] [CrossRef] [Green Version]
- Reid, D. H. 1978. Claim reserves in general insurance. Journal of the Institute of Actuaries 105: 211–315. [Google Scholar] [CrossRef]
- Renshaw, Arthur E., and Richard J. Verrall. 1998. A stochastic model underlying the chain-ladder technique. British Actuarial Journal 4: 903–23. [Google Scholar] [CrossRef]
- Ritov, Ya’acov, and Jon A. Wellner. 1988. Censoring, martingales, and the cox model. Contemporary Mathematics 80: 191–219. [Google Scholar]
- Swishchuk, Anatoliy. 2016. Change of Time Methods in Quantitative Finance. New York: Springer. [Google Scholar]
- Taylor, Greg. 2019. Loss reserving models: Granular and machine learning forms. Risks 7: 82. [Google Scholar] [CrossRef] [Green Version]
- Taylor, Greg, and Gráinne McGuire. 2016. Stochastic Loss Reserving Using Generalized Linear Models. Arlington: Casualty Actuarial Society, CAS Monograph Series; Number 3. [Google Scholar]
- Taylor, Greg, Gráinne McGuire, and James Sullivan. 2008. Individual claim loss reserving conditioned by case estimates. Annals of Actuarial Science 3: 215–56. [Google Scholar] [CrossRef]
- Taylor, Greg. 1981. Speed of finalization of claims and claims runoff analysis. ASTIN Bulletin 12: 81–100. [Google Scholar] [CrossRef] [Green Version]
- Taylor, Greg. 1982. Zehnwirth’s comments on the see-saw method: A reply. Insurance: Mathematics and Economics 1: 105–108. [Google Scholar] [CrossRef]
- Verrall, Richard J. 1991. Chain ladder and maximum likelihood. Journal of the Institute of Actuaries 118: 489–99. [Google Scholar] [CrossRef]
- Ware, James H., and David L. DeMets. 1976. Reanalysis of some baboon descent data. Biometrics 32: 459–63. [Google Scholar] [CrossRef]
- Wüthrich, Mario V. 2018. Machine learning in individual claims reserving. Scandinavian Actuarial Journal 2018: 465–80. [Google Scholar] [CrossRef]
- Zhao, Xiao Bing, and Xian Zhou. 2010. Applying copula models to individual claim loss reserving methods. Insurance: Mathematics and Economics 46: 290–99. [Google Scholar] [CrossRef]
- Zhao, Xiao Bing, Xian Zhou, and Jing Long Wang. 2009. Semiparametric model for prediction of individual claim loss reserving. Insurance: Mathematics and Economics 45: 1–8. [Google Scholar] [CrossRef]

**Figure 1.**Original data and data with unobservable delay $\tilde{T}$ cleared of operational time. The operational time in (

**b**) is estimated in Section 6. Claim counts are aggregated into monthly bins for visualization, and settlement delay is displayed in years. The red line represents the date of data collection and the green points are the date of data collection cleared of operational time effects (with respect to accident date). (

**a**) original data; (

**b**) data cleared of operational time.

**Figure 2.**Forecasting outstanding claim numbers with time-dependent development factors and chain-ladder development factors. Illustrative example with five accident years and maximum settlement delay of five years. (

**a**) forecasting with time-dependent development factors via Equation (13); (

**b**) forecasting with chain-ladder development factors via Equation (12).

**Figure 3.**Estimated components of hazard rate of the payment delay T: (

**a**) operational time estimate $\widehat{\phi}(t)$ with optimal bandwidths; (

**b**) baseline hazard estimate ${\widehat{\alpha}}_{0}(\mathcal{T}-t)$ of payment delay (in forward time) with optimal bandwidths.

**Table 1.**Estimated number of outstanding claims through hazard with operational time (op. time) and quarterly chain-ladder (CL) by accident year and payment year.

Accident Year | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | Total |

Op. Time | 0 | 0 | 0 | 0 | 0 | 23 | 92 | 171 | 254 | 513 | 1054 |

CL | 0 | 2 | 8 | 20 | 32 | 54 | 75 | 128 | 224 | 871 | 1414 |

Payment Year | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | Total |

Op. Time | 590 | 256 | 143 | 58 | 5 | 0 | 0 | 0 | 0 | 0 | 1054 |

CL | 856 | 261 | 130 | 71 | 45 | 27 | 16 | 7 | 2 | 0 | 1414 |

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Bischofberger, S.M.
In-Sample Hazard Forecasting Based on Survival Models with Operational Time. *Risks* **2020**, *8*, 3.
https://doi.org/10.3390/risks8010003

**AMA Style**

Bischofberger SM.
In-Sample Hazard Forecasting Based on Survival Models with Operational Time. *Risks*. 2020; 8(1):3.
https://doi.org/10.3390/risks8010003

**Chicago/Turabian Style**

Bischofberger, Stephan M.
2020. "In-Sample Hazard Forecasting Based on Survival Models with Operational Time" *Risks* 8, no. 1: 3.
https://doi.org/10.3390/risks8010003