THE RATES OF CONVERGENCE OF M-ESTIMATORS FOR PARTLY LINEAR MODELS IN DEPENDENT CASES

Consider the partly linear model K = X1& + go(Ti) + ei, where {(Ti, Xi)}T is a strictlystationary Sequence of random variable8, the ei’8 are i.i.d. random errorsl the K’s are realvalued responsest fo is a &vector of parameters, X is a &vector of explanatory variables,Ti is another explanatory variable ranging over a nondegenerate compact interval. Bnd ona segmnt of observations (T1, Xi 1 Y1 ),’’’ f (Tn, X;, Yn), this article investigates the rates ofconvrgence of the M-estimators for Po and go obtained from the minimisation problemwhere H is a space of B-spline functions of order m + 1 and p(-) is a function chosen suitablyUnder some regularity conditions, it is shown that the estimator of go achieves the optimalglobal rate of convergence of estimators for nonparametric regression, and the estdriator offo is asymptotically normal. The M-estimators here include regression quantile estimators,Li-estimators, Lp-norm estimators, Huber’s type M-estimators and usual least squares estimators. Applications of the asymptotic theory to testing the hypothesis H0: A’β0 =β are alsodiscussed, where β is a given vector and A is a known d × do matrix with rank d0.