Estimation of heteroscedasticity by local composite quantile regression and matrix decomposition

We propose a two-step estimation method for nonparametric model with heteroscedasticity to estimate the scale function $ \sigma (\cdot ) $ σ(⋅) and the location function $ m(\cdot ) $ m(⋅) simultaneously. The local composite quantile regression (LCQR) is employed in the first step, and a matrix decomposition method is used to estimate both $ m(\cdot ) $ m(⋅) and $ \sigma (\cdot ) $ σ(⋅) in the second step. We prove the non-crossing property of the LCQR and thereby give an algorithm, named matrix decomposition method, to ensure the non-negativity of the scale function estimator, which is much reasonable since there is no hard constraint or order adjustment to the estimators. Under some mild regularity conditions, the resulting estimator enjoys asymptotic normality. Simulation results demonstrate that a better estimator of the scale function can be obtained in terms of mean square error, no matter the error distribution is symmetric or not. Finally, a real data example is used to illustrate the proposed method.