Robust nonparametric regression with simultaneous scale curve estimation (Q1118285)

Let be given an i.i.d. sample \((X_ j\), \(Y_ j)\), \(1\leq j\leq n\), from a random vector \((X,Y)\in {\mathbb{R}}^ d\times {\mathbb{R}}\) and assume that the conditional distribution F(y\(| x)\) is of the form \(F_ 0((y- m(x))/\sigma (x))\) with a regression function m(x), a scale function \(\sigma\) (x), and an unknown distribution \(F_ 0(x)\). In analogy to Huber's simultaneous M-estimators for location and scale parameters [\textit{P. Huber}, Robust statistics. (1981; Zbl 0536.62025), section 6.4] m(x) and \(\sigma\) (x) can be defined for \(x\in {\mathbb{R}}^ d\) as unique solutions of the equations \[ (*)\quad \int \psi ((y-t)/s)dF(y| x),\quad \int \chi ((y-t)/s)dF(y| x) \] where \(\psi\) is an odd and \(\chi\) an even bounded function, satisfying some further requirements. Replacing F(y\(| x)\) by a kernel smoother \(F_ n(y| x)\), the corresponding solutions of (*) furnish M-type smoothers for m(x) and \(\sigma\) (x). Supposing various assumptions, the authors prove weak and strong consistency and asymptotic normality of these estimates and discuss their speed of convergence.