Stable distributions are well-known for their desirable properties and can effectively fit data with heavy tail. However, due to the lack of an explicit probability density function and finite second moments in most cases, traditional parametric inference methods are no longer applicable. Bayesian Synthetic Likelihood is a likelihood-free Bayesian inference method based on model simulations, which effectively addresses parameter inference problems when the probability density function is not explicitly available. Semi-parametric Bayesian Synthetic Likelihood relaxes the normality assumption by incorporating semi-parametric estimation methods, but it performs poorly when applied to data with heavy tail and excess kurtosis. To improve this, we introduced adaptive Monte Carlo algorithm to enhance the convergence speed, and transformed kernel density estimation to increase the estimation accuracy. Numerical experiments and empirical analysis on stable distributions validated the superiority of the proposed improvements.
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