Parameter Estimation Of A Partial Linear Regression Model Based On A New Symmetric Distribution

One of the major strings in developing the regression models is the assumption of the distribution of the error terms.For the sake of simplicity,it is customary to consider that the error terms follow the Normal distribution in practical applications.However,there are some drawbacks on it,such as the kurtosis of the normal distribution is three,but in many practical situations,the data distribution which is in study may be presented as platykurtic but not mesokurtic.For this reason,some scholars introduced a new symmetric distribution of low kurtosis by inserting a new parameter that a kurtosis effect depends on,which is considered as a kind of heavy-tailed distribution.The heavy-tailed distribution can well characterize this characteristic that the probability of extremum is greater than the probability of extremes in the normal distribution,so it can effectively explain the reason why extreme events exist rationally and infrequently.Because the probability density curve tail of this new distribution is thicker,which can cover more outliers.Therefore,in the analysis of low-k-variable,it makes sense to systematically study a regression model that assumes that the error terms obey the new symmetry distribution.Partially linear regression model as a combination of parametric model and nonparametric model,has the advantages of parametric model and nonparametric model,so it has more flexibility and applicability.However,since there are many heteroscedasticity and even more complicated data in real life,especially in the economic field and the test of quality improvement of industrial products.Noting the joint mean and variance model can overcome these problems because it performs both mean and variance modelling,it is able to analysis the complicated and complex data.This paper intends to establish the partially regression linear model and joint mean and variance models with new symmetric distributed errors,and using the maximum likelihood estimation and penalized spline estimation fitting parametric and nonparametric part respectively,derive the model's penalized maximum likelihood estimator and get the parameter estimators in partial linear regression model using Newton-Raphson method.Finally,the effectiveness and feasibility of the two proposed models and methods are illustrated by numerical simulation and case studies.