Abstract
This article considers estimation of regression function \(f\) in the fixed design model \(Y(x_i)=f(x_i)+ \epsilon (x_i), i=1,\ldots ,n\), by use of the Gasser and Müller kernel estimator. The point set \(\{ x_i\}_{i=1}^{n}\subset [0,1]\) constitutes the sampling design points, and \(\epsilon (x_i)\) are correlated errors. The error process \(\epsilon \) is assumed to satisfy certain regularity conditions, namely, it has exactly \(k\) (\(=\!0, 1, 2, \ldots \)) quadratic mean derivatives (q.m.d.). The quality of the estimation is measured by the mean squared error (MSE). Here the asymptotic results of the mean squared error are established. We found that the optimal bandwidth depends on the \((2k+1)\)th mixed partial derivatives of the autocovariance function along the diagonal of the unit square. Simulation results for the model of \(k\)th order integrated Brownian motion error are given in order to assess the effect of the regularity of this error process on the performance of the kernel estimator.
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References
Benhenni K (1998) Predicting integrals of stochastic processes: extensions. J Appl Probab 35:843–855
Benhenni K, Rachdi M (2005) Nonparametric estimation of the regression function from quantized observations. Comput Stat Data Anal 11(50):3067–3085
Benhenni K, Su Y (2005) Stepwise sampling schemes for estimating integrals of time series. Annales de l’I.S.U.P., XLIX, facs. 2–3, pp 19–40
Benhenni K, Rachdi M (2006) Nonparametric estimation of the average growth curve with a general non-stationary error process. C R Acad Sci Paris Sér I 343:541–544
Benhenni K, Rachdi M (2007) Nonparametric estimation of the regression function with non-stationary error process. Commun Stat Theory Methods 36(6):1173–1186
Blanke D, Vial C (2008) Assessing the number of mean square derivatives of a Gaussian process. Stoch Proc Appl 118(10):1852–1869
Boularan J, Ferré L, Vieu P (1994) Growth curves: a two-stage nonparametric approach. J Stat Plann Inference 38(3):327–350
Eugene W (2004) Stationary transformation of integrated brownian motion. arXiv:math/0412291v1, pp 18
Fan J (1992) Design-adaptive nonparametric regression. J Am Stat Assoc 87(420):998–1004
Fan J, Gijbels I (1996) Local polynomial modelling and its applications. In: Cox DR, Hinkley DV, Keiding N, Reid N (eds) Monographs on statistics and applied probability, vol 66. Chapman and Hall, London
Ferreira E, Núñez-Antón V, Rodríguez-Póo J (1997) Kernel regression estimates of growth curves using non-stationary correlated errors. Stat Probab Lett 34(4):413–423
Gasser T, Müller MG (1984) Estimating regression functions and their derivatives by the kernel method. Scand J Stat 11:171–185
Hall P, Müller HG, Wang JL (2006) Prediction in functional linear regression. Ann Stat 34:1493–1517
Härdle W (1989) Applied nonparametric regression, vol 19. Cambridge University Press, Cambridge
Hart J, Wehrly T (1986) Kernel regression estimation using repeated measurements data. J Am Stat Assoc 81:1080–1088
Lin X, Carroll RJ (2000) Nonparametric function estimation for clustered data when the predictor is measured without/with error. J Am Stat Assoc 95(450):520–534
Müller HG (1991) Smooth optimum kernel estimators near endpoints. Biometrika 78(3):521–530
Müller HG (1993) On the boundary kernel method for non-parametric curve estimation near endpoints. Scand J Stat 20:313–328
Núñez-Antón V, Woodworth G (1994) Analysis of longitudinal data with unequally spaced observations and time dependent correlated errors. Biometrics 50:445–456
Núñez-Antón V, Rodríguez J, Vieu P (1999) Longitudinal data with non-stationary errors: a nonparametric three stage approach. Test 8(1):201–231
Opsomer J, Wang Y, Yang Y (2001) Nonparametric regression with correlated errors. Stat Sci 16(2):134–153
Perrin O (1999) Quadratic variations for Gaussian processes and application to time deformation. Stoch Proc Appl 82:293–305
Rachdi M (2004) Strong consistency with rates of spectral estimation of continuous-time processes: from periodic and poisson sampling schemes. J Nonparametr Stat 16(3–4):349–364
Sarda P, Vieu P (2000) Kernel regression: Approaches, computation, and application. Wiley Series in Probability and Statistics, Ed. MG Schimek Edition, Smoothing and regression
Su Y, Cambanis S (1993) Sampling designs for estimation of a random process. Stoch Process Appl 41: 47–89
Wu O, Chiang CT, Hoover D (1998) Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. J Am Stat Assoc 93(444):1388–1402
Yao F, Müller HG, Wang JL (2005) Functional linear regression analysis for longitudinal data. Ann Stat 33:2873–2903
Zimmerman DL, Núñez-Antón V (2001) Parametric modelling of growth curve data: an overview (with comments). Test 10(1):1–73
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Benhenni, K., Rachdi, M. & Su, Y. The effect of the regularity of the error process on the performance of kernel regression estimators. Metrika 76, 765–781 (2013). https://doi.org/10.1007/s00184-012-0414-8
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DOI: https://doi.org/10.1007/s00184-012-0414-8