Estimation and test of the joint partially linear single-index model based on the skew-normal distribution

This paper constructs a joint partially linear single-index model based on the skew - normal distribution (SN-PLSIM). It integrates functions, scale, and skewness parameters to address asymmetric data modeling. We approximate the single-index functions by linear combinations of B-spline basis functions and implement parameter estimation via the Newton-Raphson iterative algorithm. A likelihood ratio test statistic is developed to distinguish between the partially linear single-index form and the linear form, and we adopt a parametric bootstrap method for practical inference. The asymptotic properties of estimators, such as consistency and rates of convergence, are theoretically established. Numerical simulations verify the estimator efficiency across different sample sizes, and real data analysis of ragweed pollen concentration shows that SN-PLSIM outperforms linear and semi - parametric models, as evaluated by AIC and likelihood ratio tests.