Alternative parameter estimation method for the Normal Inverse Gaussian Distribution

Abstract

This thesis presents a study focusing on the parameter estimation of the Normal Inverse Gaussian distribution, a specialized instance of the generalized hyperbolic distribution extensively employed in the analysis of financial time series. Traditionally, parameter estimation has relied on the maximum likelihood method and the method of moments. However, these approaches impose constraints on the feasible domain of possible skewness and excess kurtosis values. Hence, an alternative parameter estimation method for the Normal Inverse Gaussian distribution is proposed in this study, based on the Metropolis-Hastings exponential maximum likelihood method. Additionally, the efficacy of this method will be assessed by comparing it with several other estimators, including the maximum likelihood estimator, the epsilon maximum likelihood estimator, and the exponential maximum likelihood, utilizing both simulated and real-world datasets. In the simulation part, performance evaluation will be based on criteria such as the smallest root mean square error and bias, supplemented by descriptive statistics such as means and standard deviations. For the application to real-world data, model selection will be guided by a goodness-of-fit test employing the Anderson-Darling (AD) test statistic criterion. Next, the real-data is examined the autocorrelation by using a Durbin Watson (DW) statistic test. Model selection will prioritize achieving the smallest AD value alongside the highest p-value. Furthermore, the model's performance will be evaluated through the utilization of the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The method yielding the lowest AIC and BIC values is deemed optimal for parameter estimation in the Normal Inverse Gaussian distribution, particularly when analyzing bitcoin data. In the simulation results, the Metropolis-Hastings exponential maximum likelihood method consistently produced the lowest root mean square error across all scenarios. Consequently, this method is highly suitable for estimating parameters of the Normal Inverse Gaussian distribution with bitcoin data. Similarly, in practical applications, the Metropolis-Hastings exponential maximum likelihood method yielded the smallest AD value and the highest p-value, as well as the lowest AIC and BIC values. Therefore, the Metropolis-Hastings exponential maximum likelihood method proves to be the most effective approach in both simulation and application studies.