Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations

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Let {Z i; i ϵ N} be a strictly stationary real valued time series. We predict Z N + 1 from { Z 1,… Z N } by a robust nonparametric method. The predictor is defined by the kernel method and constructed as a functional M-estimate connected with the conditional law of Z p+1 on Z 1,…, Z p , when {Z i; i ϵ N} is Markovian of order p. Strong uniform convergence rates of this estimate are given together with some new results concerning robust regression kernel estimates from a sequence of R p × R valued, identically distributed and φ-mixing random pairs {( X i , Y i ); i = 1,…, n}. As a special case we obtain strong uniform convergence rates for estimators of the regression curve E( Y 1/ X 1 = ·) and of the density of the law of X 1.