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Nonparametric function estimation involving time series. (English) Zbl 0764.62038

The authors study nonparametric estimation of the conditional expectation and the conditional median of \(Y\) given \(X=x\). The observations form a stationary strongly mixing sequence \((X_ t,Y_ t)\), \(t\in{\mathbf Z}\), with \(X_ t\in{\mathbf R}^ d\) and \(Y_ t\) real-valued. Examples include nonparametric prediction in a time series \((\xi_ t)\), where \(Y_ t=\xi_{t+m}\) and \(X_ t=(\xi_ t,\xi_{t-1},\dots,\xi_{t-d+1})\). The convergence rates of local averages, and local medians, respectively, are studied. For the right choice of bandwidth, these are shown to achieve the optimal convergence rates of \(n^{-1/(2+d)}\) pointwise and in \(L_ 2\), and \((n^{-1}\log n)^{1/(2+d)}\) in \(L_ \infty\).

MSC:

62G07 Density estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G20 Asymptotic properties of nonparametric inference
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