Parameter Estimation And Hypothesis Testing Of Mode Regression Model With Complex Skew-normal Data

In the digital economy,most of the data created are skewed/asymmetrical,and extending the traditional symmetry distributions to asymmetric skewed distributions to fit such data can more accurately characterize the distributions of these skewed data.Moreover,it has been shown that the mode regression model is a powerful tool for modeling asymmetrical data.Therefore,this thesis focuses on the skew-normal mode regression model.There are still many problems that need to be solved,especially the complexity of the data structure,the optimization of the parameter estimation method,and the hypothesis testing problem.Therefore,in this thesis,we propose skew-normal mode regression model with measurement errors for covariates and use the EM algorithm to estimate the parameters.In addition,based on the skew-normal mode regression model,not only the Transform Upper-crossing/Solution algorithm(hereinafter referred to as TUS algorithm)is proposed to achieve the maximum likelihood estimation of the model parameters,but also the construction of likelihood ratio statistics and score statistics are considered to test the hypothesis of the parameters based on the estimation of the model parameters by TUS algorithm.In this thesis,the research is divided into the following three parts to address the above key issues.Firstly,the skew-normal mode regression model with measurement error data is proposed.Most of the existing studies on regression models are limited to directly observed explanatory variables,and ignoring the measurement error in the data will increase the bias in the estimation of the model parameters.Current research on measurement error models has focused on the assumption that the errors follow a normal distribution,which is not applicable to the study of asymmetrical data.Initially,the skew-normal mode regression model with measurement errors for covariates is developed,and the likelihood function of observed data is derived by integrating the conditional density function with respect to unobserved variables.At the same time,the parameters of the model are estimated using the EM algorithm by considering the complete log-likelihood function in the special case.At last,the results of the simulation study show that the mode regression of covariates with measurement error performs better than the mean regression;the use of the BMI dataset illustrates that the proposed model can be fitted to real data for analysis.Secondly,the parameter estimation method of the skew-normal mode regression model is improved.Classical parameter estimation methods,such as Gauss-Newton algorithm,EM algorithm,and two-point gradient descent algorithm,are very sensitive to the selection of initial values during iteration.Therefore,the TUS algorithm is proposed for the maximum likelihood estimation of the parameters of the skew-normal mode regression model using the concepts of a convergence algorithm developed by De Pierro(hereafter DP algorithm)and the Upper-crossing/Solution algorithm(hereafter US algorithm),which are not restricted to latent variables.The results of simulation studies and case studies show that,compared with the Gauss-Newton algorithm,the proposed TUS algorithm is much less sensitive to the initial values,more applicable,and can be well integrated with the actual data for analysis.Thirdly,the hypothesis testing problem of the skew-normal mode regression model is discussed.The homogeneity of variance is a standard assumption of the skewnormal regression model,however,this assumption is not necessarily reasonable.To begin with,the likelihood ratio statistic and the score statistic are constructed based on the derivation of the Gauss-Newton algorithm in Part II and the idea of the U-step derivation of the upper penetration function in the TUS algorithm.Then,the likelihood ratio test and the score test are used to verify the significance of the regression coefficients,the homogeneity of the scale parameters,and the homogeneity of the skewness parameters of the skew-normal mode regression model.In the end,numerical simulations and case studies yield conclusions that are consistent with the actual situation,and the case studies further illustrate that the two tests analyzed can be applied to the actual data.