Estimation and Test of the Joint Partially Linear Single-Index Model Based on the Skew-Normal Distribution

Based on the proposed model, this paper first approximates the single-index functions in the model using the linear expansion of B-spline basis functions, and thoroughly investigates the parameter estimation of the joint partially linear single-index model for the location, scale, and skewness parameters under the skew-normal distribution, with the corresponding estimation algorithm developed. Next, given the complexity of the single-index model, we construct a likelihood ratio test statistic to examine whether the model follows a partially linear single-index specification or a linear specification, and propose a parametric bootstrap procedure for the test. Furthermore, to theoretically justify the performance of the proposed estimators, we rigorously derive the relevant asymptotic properties of the joint partially linear single-index model, including the consistency and convergence rates of the estimators. In addition, we conduct an extensive numerical simulation study to evaluate the finite-sample performance of the proposed parameter estimation method and the likelihood ratio test statistic under various sample sizes. Finally, we apply the joint partially linear single-index model to the ragweed pollen concentration data, and compare it with several competing linear and semiparametric models. Using the model selection criterion AIC and the likelihood ratio test, we demonstrate that the joint partially linear single-index model for the location, scale, and skewness parameters based on the skew-normal distribution outperforms its competing models in terms of goodness of fit.