This study proposes a nonstationary Poisson-Lindley Hidden Markov Model (PL-HMM) as a novel framework for modeling the frequency of community-based disaster insurance claims. The model accounts for both serial dependence and overdispersion in claim counts through hidden risk states, while nonstationary transition probabilities are introduced via a sliding-window mechanism. Parameters are estimated using the Generalized Expectation-Maximization (GEM) algorithm, supported by a theoretical foundation to ensure a monotonic improvement of the complete log-likelihood. The model was simulated using monthly claim frequency data from West Java Province, Indonesia. A comparative analysis against nonstationary Poisson HMMs with varying numbers of hidden states showed that the two-state nonstationary PL-HMM achieved the lowest Bayesian information criterion ($ BIC $), thus indicating the best fit. A sensitivity analysis of sliding-window horizons (12, 24, and 36 months) demonstrated that persistence patterns of claim risk-states remained robust, with horizon changes reflecting alternative risk measurement periods. The results highlight that the proposed model effectively captures time-varying claim risks, particularly the alternation between low- and high-claim periods, while realistically reflecting the empirical dominance of high-claim regimes. Beyond the simulation data, a nonstationary PL-HMM is flexible and applicable to other regions that exhibit overdispersed claim data, making it a valuable framework for adaptive premium design and disaster risk financing in community-based insurance schemes.
Citation: Hilda Azkiyah Surya, Sukono, Herlina Napitupulu, Noriszura Ismail. Nonstationary transition Poisson-Lindley Hidden Markov model for community-based disaster insurance claim[J]. AIMS Mathematics, 2025, 10(10): 23411-23428. doi: 10.3934/math.20251040
This study proposes a nonstationary Poisson-Lindley Hidden Markov Model (PL-HMM) as a novel framework for modeling the frequency of community-based disaster insurance claims. The model accounts for both serial dependence and overdispersion in claim counts through hidden risk states, while nonstationary transition probabilities are introduced via a sliding-window mechanism. Parameters are estimated using the Generalized Expectation-Maximization (GEM) algorithm, supported by a theoretical foundation to ensure a monotonic improvement of the complete log-likelihood. The model was simulated using monthly claim frequency data from West Java Province, Indonesia. A comparative analysis against nonstationary Poisson HMMs with varying numbers of hidden states showed that the two-state nonstationary PL-HMM achieved the lowest Bayesian information criterion ($ BIC $), thus indicating the best fit. A sensitivity analysis of sliding-window horizons (12, 24, and 36 months) demonstrated that persistence patterns of claim risk-states remained robust, with horizon changes reflecting alternative risk measurement periods. The results highlight that the proposed model effectively captures time-varying claim risks, particularly the alternation between low- and high-claim periods, while realistically reflecting the empirical dominance of high-claim regimes. Beyond the simulation data, a nonstationary PL-HMM is flexible and applicable to other regions that exhibit overdispersed claim data, making it a valuable framework for adaptive premium design and disaster risk financing in community-based insurance schemes.
| [1] |
D. Lefutso, A. Ogundeji, G. Danso-Abbeam, Y. Nyam, Low-income households' willingness to pay for flood risk insurance in South Africa, Progress in Disaster Science, 25 (2025), 100403. https://doi.org/10.1016/j.pdisas.2024.100403 doi: 10.1016/j.pdisas.2024.100403
|
| [2] |
I. Azis, B. Setiawaty, I. Gusti Putu Purnaba, Modeling of vehicle insurance claim using exponential hidden Markov, International Journal of Pure and Applied Mathematics, 118 (2018), 309–320. https://doi.org/10.12732/ijpam.v118i2.15 doi: 10.12732/ijpam.v118i2.15
|
| [3] |
Z. Oflaz, C. Yozgatligil, A. Selcuk-Kestel, Aggregate claim estimation using bivariate hidden markov model, ASTIN Bulletin: The Journal of the IAA, 49 (2019), 189–215. https://doi.org/10.1017/asb.2018.29 doi: 10.1017/asb.2018.29
|
| [4] |
R. Elliott, T. Siu, Hedging options in a hidden Markov-switching local-volatility model via stochastic flows and a Monte-Carlo method, J. Futures Markets, 43 (2023), 925–950. https://doi.org/10.1002/fut.22422 doi: 10.1002/fut.22422
|
| [5] |
L. Slater, B. Anderson, M. Buechel, S. Dadson, S. Han, S. Harrigan, et al., Nonstationary weather and water extremes: a review of methods for their detection, attribution, and management, Hydrol. Earth Syst. Sci., 25 (2021), 3897–3935. https://doi.org/10.5194/hess-25-3897-2021 doi: 10.5194/hess-25-3897-2021
|
| [6] |
A. Tsoi, S. Zhang, M. Hagenbuchner, Pattern discovery on Australian medical claim data-a systematic approach, IEEE Trans. Knowl. Data Eng., 17 (2005), 1420–1435. https://doi.org/10.1109/TKDE.2005.168 doi: 10.1109/TKDE.2005.168
|
| [7] |
V. Koerniawan, N. Sunusi, R. Raupong, Estimasi parameter model Poisson hidden Markov pada data Banyaknya Kedatangan Klaim Asuransi Jiwa, ESTIMASI: Journal of Statistics and Its Application, 1 (2020), 65–73. https://doi.org/10.20956/ejsa.v1i2.9302 doi: 10.20956/ejsa.v1i2.9302
|
| [8] |
W. Al-Nuaami, A. Heydari, H. Khamnei, The Poisson-Lindley distribution: some characteristics, with its application to SPC, Mathematics, 11 (2023), 2428. https://doi.org/10.3390/math11112428 doi: 10.3390/math11112428
|
| [9] |
N. Azizah, S. Astutik, Nurjannah, Two-state Poisson hidden Markov models for analysis of seismicity activity rates in west Nusa Tenggara, IOP Conf. Ser.: Mater. Sci. Eng., 546 (2019), 052015. https://doi.org/10.1088/1757-899X/546/5/052015 doi: 10.1088/1757-899X/546/5/052015
|
| [10] |
K. Orfanogiannaki, D. Karlis, G. Papadopoulos, Identification of temporal patterns in the seismicity of Sumatra using Poisson Hidden Markov models, Research in Geophysics, 4 (2014), 4969. https://doi.org/10.4081/rg.2014.4969 doi: 10.4081/rg.2014.4969
|
| [11] | R. Paroli, G. Redaelli, L. Spezia, Poisson hidden Markov models for time series of overdispersed insurance counts, Proceedings of XXXI Astin Colloqium, 2000,461–474. |
| [12] |
S. Chatterjee, O. Romero, S. Pequito, Analysis of a generalised expectation-maximisation algorithm for Gaussian mixture models: a control systems perspective, Int. J. Control, 95 (2022), 2734–2742. https://doi.org/10.1080/00207179.2021.1931964 doi: 10.1080/00207179.2021.1931964
|
| [13] | Kedeputian bidang sistem dan strategidirektorat pemetaan dan evaluasi risiko bencana, Kajian rsiko bncana nsional povinsi Jawa Barat 2022–2026, Badan Nasional Penanggulangan Bencana, 2021. Available from: https://inarisk.bnpb.go.id/pdf/Jawa%20Barat/Dokumen%20KRB%20Prov.%20Jawa%20Barat_final%20draft.pdf. |
| [14] |
H. Surya, Sukono, H. Napitupulu, N. Ismail, A systematic literature review of insurance claims risk measurement using the hidden Markov model, Risks, 12 (2024), 196. https://doi.org/10.3390/risks12110169 doi: 10.3390/risks12110169
|
Figures(4) / Tables(8)
Hilda Azkiyah Surya, Sukono, Herlina Napitupulu, Noriszura Ismail. Nonstationary transition Poisson-Lindley Hidden Markov model for community-based disaster insurance claim[J]. AIMS Mathematics, 2025, 10(10): 23411-23428. doi: 10.3934/math.20251040
DownLoad: