Local Sparse Estimation Of Functional Quantile Regression Model

Functional data is a complex data type in many fields,including economics,meteorology,medicine,and more.Therefore,researching functional regression models has important practical significance.However,the assumption of normal distribution of error terms in traditional functional linear regression models limits their scope of application.To address this limitation,this article studies functional quantile regression models,which can handle situations where error terms have skewed and heavy tailed distributions,thus improving the fitting performance of the data.The first part primarily employs Bayesian algorithms and EM algorithms for parameter estimation of functional quantile regression models,and derives the estimation process of coefficient functions.In the Bayesian algorithm,a suitable smooth spline prior is adopted,and the Gibbs sampling is applied to estimate the parameters.Since there are latent variables in the model,the EM algorithm is used for iterative estimation by constructing the full likelihood function and applying smooth spline penalties until the coefficients converge to the estimated values.Numerical simulation results show that functional quantile regression models are more robust than functional linear regression models.Furthermore,compared with the Bayesian algorithm,the functional quantile regression model estimated by the EM algorithm has better fitting performance,higher accuracy,and significantly improved parameter estimation as the sample size increases.Finally,this study applies the functional quantile regression model estimated by the EM algorithm to empirical analysis of the data of the top 100 stocks on the Shanghai and Shenzhen Stock Exchanges,and investigates the relationship between daily stock prices and peak prices.The research results show that January to March is a suitable time to sell stocks.The main focus of the second part is to effectively identify empty subregions and construct objective functions by applying SCAD penalty terms.Due to the excellent performance of the EM algorithm in parameter estimation of functional quantile regression models,a smooth estimate is generated by iterative EM steps,achieving local sparse estimation.When this method is applied to numerical simulations,the results indicate its superiority;local sparse estimation of functional quantile regression models has smaller errors in coefficient functions for empty subregions,verifying the validity of the model.In the empirical analysis,the relationship between dailyPM₁₀ and air quality index is studied,and the effect of local sparse estimation is evident.Changing the quantile point yields different clusters of curves at different quantile levels,effectively inferring the actual range of air quality index values at different time points.