ESTIMASI BAYESIAN PADA PARAMETER HUKUM MOTALITA GOMPERTZ MENGGUNAKAN ALGORITMA METROPOLIS-HASTINGS
Abstract
Tingkat mortalitas merupakan salah satu hal penting untuk menentukan nilai premi pada suatu produk asuransi. Pada umumnya, perusahaan asuransi menggunakan tabel mortalitas deterministik yang dibangun dari data kematian masa lalu. Namun pada kenyataannya, tingkat mortalitas dipengaruhi oleh faktor-faktor ketidakpastian yang menyebabkan tingkat mortalitas tersebut berubah secara stokastik. Pada penelitian ini, akan dikaji pengaruh mortalitas stokastik dalam mengestimasi parameter hukum mortalitas Gompertz menggunakan pendekatan analisis Bayesian sehingga parameter-parameter pada hukum mortalitas Gompertz tidak lagi berbentuk konstanta, namun memiliki distribusi. Estimasi Bayesian dilakukan dengan asumsi distribusi prior adalah normal. Pelibatan unsur stokasik dilakukan dengan menambahkan gangguan mortalitas yang dinyatakan dalam persentase dari force of mortality dengan rentang . Simulasi numerik dilakukan menggunakan Markov Chain Monte Carlo (MCMC) dengan Algoritma Mettopolis-Hastings. Hasil simulasi menunjukkan bahwa dengan l0 = 100000 dan l111 = 0, diperoleh nilai m* berdistribusi normal dengan mean 0.001665164 dan variansi 9,525 × 10-9 dan C* berdistribusi normal dengan mean 1.081264461 dan variansi 6,312134 × 10-7. Hasil ini dapat digunakan sebagai kerangka kerja yang lebih akurat untuk menganalisis keandalan, ketahanan, dan pembiayaan dalam dunia aktuaria, serta memberikan dasar yang lebih baik untuk pengelolaan risiko perusahaan.
Abstract
The mortality rate is a crucial factor in determining the premium value for an insurance product. Typically, insurance companies use deterministic mortality tables that are built from past death data. However, in reality, the mortality rate is influenced by various uncertainty factors that cause it to change stochastically. In this research, we will study the influence of stochastic mortality in estimating the parameters of the Gompertz mortality law using a Bayesian analysis approach. This will enable us to model the parameters in the Gompertz mortality law as a distribution rather than a constant value. Bayesian estimation is carried out assuming the prior distribution is normal. The involvement of stochastic elements is carried out by adding mortality disturbance ∈ which is expressed as a percentage of the force of mortality with a range of . Numerical simulations were carried out using Markov Chain Monte Carlo (MCMC) with the Mettopolis-Hastings Algorithm. The simulation results show that with l0 = 100000 and l111 = 0, the m* value is normally distributed with a mean of 0.001665164 dan and a variance of 9,525 × 10-9 and C* is normally distributed with a mean of 1.081264461 and a variance of 6,312134 × 10-7. These results can be used as a more accurate framework for analyzing reliability, resilience and financing in the actuarial world, as well as providing a better basis for enterprise risk management.
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DOI: http://dx.doi.org/10.30821/axiom.v12i2.18061
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