Model Selection Of Single-index Model With Heredity Constraint

The single-index model is a widely used semiparametric regression model.It has good interpretability of the parametric model,and also has the flexibility of the nonparametric model.What's more,it avoids the curse of dimensionality of high-dimensional nonparametric.In this paper,we consider the variable selection and parameter estimation of the single-index model with heredity constraint.Compared with the classical single-index model,we not only study the influence of each predictor on the response variable,but also consider the interaction between variables in model.The heredity constraint condition means only when the model exists the main effect,the interaction effect may exist;on the contrary,if an interaction effect exists,the corresponding main effect must exists.To make the model satisfy the condition of heredity constraints,we redefine the parametric model.The estimation based on penalty variable selection method has the properties of sparsity,consistency,and asymptotic normality.It means that the paper proposed estimator has the oracle asymptotic property and meets the heredity constraints conditions.Simulation studies show that the proposed estimation method performs well under finite samples.The rest of the paper is organized as follows.In section 1,we introduce the background of the single-index model with heredity constraint,the current research results and the main problems solved in the paper.The method of single-index model selection with heredity constraint and the implementation procedure about the estimation method of parameter coefficient are studied in section 2.The asymptotic theory results about the coefficient estimation of single-index model with heredity constraint are discussed in section 3.In section 4,we verify the parameter estimation method is effective by using numerical simulation.The detailed proof process of the theorem under the assumption condition are given in section 5.