Bahadur Representation for the Nonparametric M-Estimator Under Alpha-mixing Dependence

Under the condition that the observations, which come from a high-dimensional population (<I>X,Y</I>), are strongly stationary and strongly-mixing, through using the local linear method, we investigate, in this paper, the strong Bahadur representation of the nonparametric <I>M</I>-estimator for the unknown function <I>m(x)</I>=arg min<SUB>a</SUB><I>IE(<font face="Symbol">r</font>(a,Y)</I>|<I>X=x)</I>, where the loss function <I><font face="Symbol">r</font>(a,y)</I> is measurable. <BR>Furthermore, some related simulations are illustrated by using the cross validation method for both bivariate linear and bivariate nonlinear time series contaminated by heavy-tailed errors. The <I>M</I>-estimator is applied to a series of S&P 500 index futures andspot prices to compare its performance in practice with the "usual" squared-loss regression estimator.