On consistency for wavelet estimator of regression function based on biased samples under extended negatively dependence

For a nonparametric regression model with biased samples under extended negatively dependence, we concern with the estimation problem of the regression function over Sobolev space. On the basis of the wavelet kernel, we propose a wavelet estimator and investigate its consistency properties under mild conditions. For the proposed wavelet estimator, we establish the pth mean consistency based on moment inequality and get the complete consistency by virtue of exponential inequality and truncation technique. It should be pointed out that the complete consistency of wavelet estimator is a novel theoretical result in the nonparametric regression model with biased samples. In addition, the effectiveness of the proposed wavelet estimator is demonstrated on some numerical experiments.