Composite Expectile Estimation in Partial Functional Linear Regression Model

Recent research and substantive studies have shown growing interest in expectile regression (ER) procedures. Similar to quantile regression, ER with respect to different expectile levels can provide a comprehensive picture of the conditional distribution of a response variable given predictors. This study proposes three composite-type ER estimators to improve estimation accuracy. The proposed ER estimators include the composite estimator, which minimizes the composite expectile objective function across expectiles; the weighted expectile average estimator, which takes the weighted average of expectile-specific estimators; and the weighted composite estimator, which minimizes the weighted composite expectile objective function across expectiles. Under certain regularity conditions, we derive the convergence rate of the slope function, obtain the mean squared prediction error, and establish the asymptotic normality of the slope vector. Simulations are conducted to assess the empirical performances of various estimators. An application to the analysis of capital bike share data is presented. The numerical evidence endorses our theoretical results and confirm the superiority of the composite-type ER estimators to the conventional least squares and single ER estimators.