Bahadur representation for the nonparametric M-estimator under α-mixing dependence

Under the condition that the observations, which come from a high-dimensional population (X, Y), are strongly stationary and strongly mixing, through using the local linear method, we investigate in this paper, the strong Bahadur representation of the nonparametric M-estimator for the unknown function m(x)=arg min a(ρ (a, Y)|X=x), where the loss function ρ(a, y) is measurable. 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 M-estimator is applied to a series of S&P 500 index futures and spot prices to compare its performance in practice with the 'usual' squared-loss regression estimator.
Keywords: asymptotic representation; kernel function; robust estimator; strongly mixing