Numerical characteristics and parameter estimation of finite mixed generalized normal distribution

Abstract

In this paper, a univariate finite mixed generalized normal distribution (MixGND) is proposed. First, we derive some probabilistic properties including hazard rate function, characteristic function, kurtosis and skewness, for a mixture of two generalized normal distributions. In particular, we use a geometric analysis and numerical simulation technique to study the monotonicity of skewness and kurtosis from prescribing corresponding parameters. Then moment estimation and maximum likelihood estimation of parameters are also given. To use the maximum likelihood estimation (MLE) method, an expectation conditional maximization (ECM) algorithm is proposed to estimate and numerically simulate seven parameters of a two-component MixGND under the same variance and heteroscedasticity. By using data sets of the S&P 500 and Shanghai Stock Exchange Composite Index (SSEC), we compare goodness-of-fit performance between the mixture of two generalized normal distributions and the mixture of two normal distributions. The empirical analysis results show that the former better describes the heavy-tailed and leptokurtic characteristics of the daily returns.

Keywords:

• ECM algorithm
• Maximum likelihood estimation
• Mixed generalized normal distribution
• Numerical characteristics

Acknowledgments

We are thankful to the Editorial Board and to the reviewers for their valuable comments and suggestions, which helped us to greatly improve the manuscript.

Disclosure statement

There are no conflicts of interest regarding the publication of this paper.

Additional information

Funding

This research is supported by the National Science Foundation of China (grant 61573006).