# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bierens, H.J., Uniform consistency of kernel estimators of a regression function under generalized conditions, J. amer. statist. assoc., 78, 699-707, (1983) · Zbl 0565.62027 [2] Collomb, G.; Härdle, W., Strong uniform convergence rates in robust nonparametric time series analysis, Stochastic process. appl., 23, 77-89, (1986) · Zbl 0612.62127 [3] Cox, D.R., Long-range dependence: A review, (), 55-74 [4] Farrell, R.H., On the lack of a uniformly consistent sequence of estimators of a density function in certain cases, Ann. math. statist., 38, 471-474, (1967) · Zbl 0158.17804 [5] Farrell, R.H., On the best obtainable asymptotic rates of convergence in estimation of a density function at a point, Ann. math. statist., 43, 170-180, (1972) · Zbl 0238.62049 [6] Granger, C.W.J.; Joyeaux, R., An introduction to long-memory time series models and fractional differencing, J. time series anal., 1, 15-29, (1980) · Zbl 0503.62079 [7] Grenander, U.; Szegö, G., Toeplitz forms and their applications, (1958), University of California Press · Zbl 0080.09501 [8] Hall, P.; Hart, J.D., Convergence rates in density estimation for data from infinite-order moving average processes, Probab. theory rel. fields, (1990), to appear · Zbl 0695.60043 [9] Hart, J.D., Kernel smoothing when the observations are correlated, () [10] Hart, J.D., Kernel regression estimation with time series errors, J. roy. statist. soc. ser. B, (1990), to appear [11] Mandelbrot, B.B.; Van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM rev., 10, 422-437, (1968) · Zbl 0179.47801 [12] Priestley, M.B., Spectral analysis and time series, (1981), Academic Press New York · Zbl 0537.62075 [13] Truong, Y.K.; Stone, C., Nonparametric time series prediction: kernel estimates based on local averages, (1988), unpublished manuscript [14] Zygmund, A., Trigonometric series, Vol. 1, (1959), Cambridge University Press · JFM 58.0280.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.