Testing the Heteroscedasticity in Single-Index Model

In the additive regression models, the single-index model is considered commonly for high dimensional regression analysis. The specification of this model that it is more flexible compared with a parametric model, and it avoids the curse of dimensionality because the single-index reduces the dimensionality of a standard variable vector (x in the multi-regression) to a univariate index (\beta^\T X in the single-index model). In this paper, we developed a single-index regression model with a functional errors' term that serves in checking the heteroscedasticity. Since the efficient inference of a regression model demands that heteroscedasticity is regarded when it exists, this paper presents the assumptions of testing variance constancy in single-index models. The test statistic is assessing the variance homogeneity stated as a combination of Levene's test and the theories of ANOVA for the infinite factor levels. The test statistic in the simulation studies displays appropriately in all situations compared to a well-known method and applies to a real dataset.