Bootstrap-Based Initialization for Maximum Likelihood Estimation in The Infinite Mixture Distributions and Its Application to Automobile Insurance Claims

When applying maximum likelihood estimation to the infinite mixture distributions, closed-form expressions for the parameters are rarely obtainable, so numerical methods, such as the Newton–Raphson technique, are often employed. A primary challenge in numerical methods is selecting suitable initial values. In this paper, the bootstrap approach is applied to determine initial parameter values for maximum likelihood estimation in the infinite mixture distributions. The bootstrap method is employed to generate the mixing distribution. The parameter estimates of the mixing distribution are used as initial values for performing maximum likelihood estimation on the infinite mixture distributions. In this study, both nonparametric and parametric bootstrap approaches are applied. Monte Carlo simulations are used to assess the performance of both bootstrap approaches. Simulation results indicate that the method of moments and the two bootstrap-based approaches yield identical maximum likelihood estimators. The study also reveals that when raw moments are unavailable or undefined, both bootstrap-based methods, especially the nonparametric bootstrap, offer a reliable means of determining initial values. The proposed method showed good performance when applied on the third-party liability claims data in Indonesia.