Consistency of Wavelet Estimators for a Family of Regression Functions

Smoothness of the regression function is unknown in practical applications, so it isn't reasonable to assume the smoothness. This paper investigates a regression model based on size-biased random samples, and proves the mean consistency of two wavelet estimators without assuming any smooth conditions of the regression function. Numerical experiments indicate that regression functions given by the two wavelet estimators are closer to the true regression function as the size of sample increasing. Especially, they are almost same as the true regression function when the size of sample is 1 000. Numerical experiments support the main theorem. The results can be considered as a supplement of Chesneau and Shirazi's work.