Statistical Inference Of Mixture Regression Model With Single-Index Varying-Coefficient

The emergence of the Mixture Model has a long history. Due to the flexibility to fit local volatility in the complex data, for instance, multi-modality, it has gradually been an important tool that can mine information inside of data, and applied in many fields. Early,restricted by the immaturity of computational methods of parameter esti-mation,it was in a period of slow development. In 1977, EM algorithm was officially put forward. This iterative algorithm conform the idea of maximum likelihood, and has the advantages of simple feasibility and stable convergence. Gradually, it becomes the primary algorithm for computing parameter estimation of Mixture model. More-over, the ideas of studying the relation between dependent variable and independent variables introduced, its researches step into a brand new stage again. In addition,the identification, the number of the mixture layer and label switching are also the concerns of researching such a model.Limited by curse of dimensionality, the proposed non-parametric or semi-parametric mixture models just used an one-dimensional covariate. Inspired by single-index regression model and mixture of varying-coefficient regression model, we pro-pose a novel model that can use multi-dimensional covariates without worrying about curse of dimensionality. The main work of this paper is as follows. Firstly, we pro-pose the novel mixture of single-index varying-coefficient regression model and prove the existence of identifiability of it. Then, we give out the EM algorithm to estimate the non-parametric functions ?(u) and index coefficient a. More specifically, we use kernel function to estimate local constant values of 0(u),and use linear interpola-tion to ensure that the values are continuous and smooth in each iteration. For index coefficient estimation of ?, we use the grid method to update it. We use a 10-fold cross-validation method to determine the optimal bandwidth. Then, under appropriate regular conditions, we present the asymptotic properties of the estimators and their corresponding proofs. Finally, by simulation experiments of comparing with other models, we show the advantages of the model in this paper.