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Nonparametric regression with long-range dependence. (English) Zbl 0713.62048
Summary: The effect of dependent errors in fixed-design, nonparametric regression is investigated. It is shown that convergence rates for a regression mean estimator under the assumption of independent errors are maintained in the presence of stationary dependent errors, if and only if \(\sum r(j)<\infty\), where r is the covariance function. Convergence rates when \(\sum r(j)=\infty\) are also investigated.
In particular, when the sample is of size n, when the mean function has k derivatives and \(r(j)\sim C| j|^{-\alpha}\), the rate is \[ n^{-k\alpha /(2k+\alpha)}\text{ for } 0<\alpha <1\text{ and } (n^{-1} \log n)^{k/(2k+1)}\text{ for } \alpha =1. \] These results refer to optimal convergence rates. It is shown that the optimal rates are achieved by kernel estimators.

MSC:
62G07 Density estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G20 Asymptotic properties of nonparametric inference
62J02 General nonlinear regression
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