Heteroscedasticity Test And Variance Estimation For Single Index Model

Parametric regression models are the most commonly used models in statistical analysis.However,there are many situations in real life where it is difficult to apply simple parametric regression models.As a result,people have proposed non-parametric regression models.Compared to parametric regression models,such models have looser assumptions about the distribution of variables and broadened the scope of application.The semi-parametric regression model is between parametric and non-parametric.It combines the advantages of the two types of models and has good properties.The single index model is a common semi-parametric regression model with index terms.This model can not only achieve the effect of data dimensionality reduction,but also maintain the advantages of non-parametric smoothness.It has been widely used in economics,agriculture,medicine and other fields.The independent homoscedasticity of the random error term is a basic assumption in the regression model.If this assumption is not true,it will lead to many problems.For example,common parameter estimation methods are no longer effective,and hypothesis testing is meaningless,which will affect the accuracy of statistical inference.Therefore,it is important to perform heteroscedasticity tests on models.At present,many scholars have conducted related research on heteroscedasticity problems and proposed many effective testing methods.However,relatively few studies have been conducted on heteroscedasticity tests for single-index models.Considering the importance of the heteroscedasticity test to the statistical inference of the regression model and the unique properties of the single-index model,this paper proposes a new test statistic based on the estimation equation method,combined with the completely non-parametric variance function test method to test the heteroscedasticity of the single-index model.Monte Carlo simulation and case analysis are used by RStudio to calculate the experience level and experience power of the test statistics.The experimental results show that the combination of the estimation equation method and the completely non-parametric variance function method can effectively test the model's heteroscedasticity.With the development of the times,the data generated in many fields are gradually high-dimensional,which has brought unprecedented challenges to statistical research.In addition,considering the estimation of variance is a basic part of statistical modeling.The problem is also an indispensable step in the process of statistical inference,and it is also important for the problem of variable selection.This paper combines the block 3x2 cross-validation method with three effective variable selection methods:SIS,LASSO,and SCAD to estimate the variance of a single-index model in high-dimensional situations.And the estimated equation method is used to estimate the unknown parameters and unknown link functions in the model.Finally,the Monte Carlo simulation was performed by RStudio,and compared with the naive two-stage method and the refitted cross-validation method.The experimental results show that the block 3×2 cross-validation method is more robust in the variance estimation process.