Statistical Inference For Nonlinear Joint Mean And Variance Models With Mixture Data

With the rapid development of of science and technology, not only makes people’s lives have a deeper understanding of the environment, but also makes people’s quality of life requirements increasing. It also makes all kinds of data aggregation from real life becomes very huge, which lead us to deal with these data becomes more complex. When we need to study these complex real data, if we make a simple analysis of the overall data, it is difficult to find the difference between the various types of data. In order to better analysis of the data, the statisticians often used cluster analysis, the overall data are classified according to different indicators or characteristics, then the data with similar properties or similar index are analyzed in detail. Mixture regression models are one of the most important statistical data analysis tools in a heterogeneous population which main research contains two or more than two children of mixed data clustering, mixture regression models have been applied in various fields including biology, medicine, economics, environmental science, sampling survey, engineering technology and so on. The existence of a large number of different variance data break the traditional regression model of China’s poor assumptions, In order to effectively control the variance, understand the sources of variance, it is necessary to model the variance, joint mean and variance model is one of the most important research tools to process heteroscedasticity. There is always a nonlinear relationship between the variables. As a result, it also makes a simple linear regression model whichn has been widely used model in many areas, does not see more in the application of the actual problem. Although some nonlinear models can transform it into linear models, in fact, more nonlinear models can not be transformed into a linear model.In this paper, based on heterogeneous population, mixture data, heteroscedasticity and non-linear models, we investigate the maximum likelihood estimate for unknown parameters based on Expectation Maximization (EM) algorithm with normal distribution and skew normal distribution. The main content includes the following sections:Firstly, established nonlinear model for the parameters of joint mean and variance of the models. Based on mixture normal distribution data, we propose mixture of nonlinear for joint mean and variance models in this part. Then we investigate the maximum likelihood estimate for unknown parameters based on Expectation Maximization (EM) algorithm and give the formulas which the Expectation Maximization (EM) algorithm need. Furthermore, we make some simulations to show that the proposed procedure works satisfactorily. Lastly, a real example is presented to illustrate the proposed methodology.Secondly, the normal distribution data is symmetrical, but the real data, such as finance, economics, biomedical science, environmental science and many other fields are nearly normal, they are not strictly symmetrical and have certain deviation. From the skew normal distribution’s property, we know that the skew normal distribution will be normal distribution when the skewness is zero. So the normal distribution is a special case of the skew normal distribution. Therefore, on the basis of the first part of the study, we proposed the nonlinear regression models and the nonlinear for joint location and scale models with mixture skew normal distribution data in this part and investigated the maximum likelihood estimate for unknown parameters based on the Expectation Maximization (EM) algorithm. Furthermore, we make some simulations to show that the proposed procedure works satisfactorily.Finally, on the basis of the second part, we made a nonlinear model for the skewness parameter and proposed mixture of nonlinear for joint location, scale and skewness models with mixture skew normal distribution data in this part. Then we investigated the maximum likelihood estimate for unknown parameters based on Expectation Maximization (EM) algorithm and some simulations to show that the proposed procedure works satisfactorily.