On asymptotic expansion of pseudovalues in nonparametric median regression

Summary

We consider the median regression model X_k = θ(x_k) + ξ_k, where the unknown signal θ: [0,1] → ℝ, is assumed to belong to a Hölder smoothness class, the ξ_ks are independent, but not necessarily identically distributed, noises with zero median. The distribution of the noise is assumed to be unknown and satisfying some weak conditions. Possible noise distributions may have heavy tails, so that, for example, the expectation of noises does not exist. This implies that in general linear methods (for example, kernel method) cannot be applied directly in this situation. On the basis of a preliminary recursive estimator, we construct certain variables Y_ks, called pseudovalues which do not depend on the noise distribution, and derive an asymptotic expansion (uniform over a certain class of noise distributions): Y_k = θ(x_k) + ∊_k + r_k, where ∊_ks are binary random variables and the remainder terms r_ks are negligible. This expansion mimics the nonparametric regression model with binary noises. In so doing, we reduce our original observation model with “bad” (heavy-tailed) noises effectively to the nonparametric regression model with binary noises.