## Electronic Journal of Statistics

#### Abstract

We propose an adaptive smoothing method for nonparametric regression. The central idea of the proposed method is to “calibrate” the estimated function through an adaptive bandwidth function, which is a kind of intermediate solution between the global bandwidth (constant on the support) and the local bandwidth (variable with $x$). This also allows to correct the bias of the local polynomial estimator, with some benefits for the inference based on such estimators. Our method, which uses the Neural Network technique in a preliminary (pilot) stage, is based on a rolling, plug-in, bandwidth selection procedure. It automatically reaches a trade-off between the efficiency of global smoothing and the adaptability of local smoothing. The consistency and the optimal convergence rate of the resulting bandwidth estimators are shown theoretically. A simulation study shows the performance of our method for finite sample size.

#### Article information

Source
Electron. J. Statist., Volume 8, Number 2 (2014), 2848-2878.

Dates
First available in Project Euclid: 8 January 2015

https://projecteuclid.org/euclid.ejs/1420726193

Digital Object Identifier
doi:10.1214/14-EJS966

Mathematical Reviews number (MathSciNet)
MR3299124

Zentralblatt MATH identifier
1308.62083

#### Citation

Giordano, Francesco; Parrella, Maria Lucia. Global adaptive smoothing regression. Electron. J. Statist. 8 (2014), no. 2, 2848--2878. doi:10.1214/14-EJS966. https://projecteuclid.org/euclid.ejs/1420726193

#### References

• Barron, A. R. (1993). “Universal approximation bounds for superpositions of a sigmoidal function”, IEEE Transactions on Information Theory, 39, 930–945.
• Barron, A. R. (1994). “Approximation and estimation bounds for artificial neural networks”, Machine Learning, 14, 115–133.
• Cao, R. (2001). “Relative efficiency of local bandwidths in kernel density estimation”, Statistics, 35, 113–137.
• Chen, X. and Shen, X. (1998). “Sieve extremum estimates for weakly dependent data”, Econometrica, 66, 289–314.
• Choi, E., Hall, P. and Rousson, V. (2000). “Data sharpening methods for bias reduction in nonparametric regression”, Annals of Statistics, 28, 1339–1355.
• Fan, J. and Gijbels, I. (1995). “Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation”, Journal of the Royal Statistical Society, series B, 57, 371–394.
• — (1996)., Local Polynomial Modelling and Its Applications, Chapman and Hall, London.
• Fan, J., Gijbels, I., Hu, T.-C. and Huang, L.-S. (1996). “A study of variable bandwidth selection for local polynomial regression”, Statistica Sinica, 6, 113–127.
• Fan, J. and Huang, L.-S. (1999). “Rates of convergence for the pre-asymptotic substitution bandwidth selector”, Statistics and Probability Letters, 43, 309-316.
• Gao, J. and Gijbels, I. (2008). “Bandwidth selection in nonparametric kernel testing”, Journal of the American Statistical Association, 103, 1584–1594.
• Giordano, F. and Parrella, M. L. (2008). “Neural networks for bandwidth selection in local linear regression of time series”, Computational Statistics & Data Analysis, 52, 2435–2450.
• Gluhovsky, I. and Gluhovsky, A. (2007). “Smooth location-dependent bandwidth selection for local polynomial regression”, Journal of the American Statistical Association, 102, 718–725.
• Hall, P. and Schucany, W. R. (1989). “A local cross-validation algorithm”, Statistics and Probability Letters, 8, 109–117.
• Härdle, W., Hall, P. and Marron, J. S. (1988). “How far are automatically chosen regression smoothing parameters from their optimum?”, Journal of the American Statistical Association, 83, 86–99.
• He, H. and Huang, L.-S. (2009). “Double-smoothing for bias reduction in local linear regression”, Journal of Statistical Planning and Inference, 139, 1056–1072.
• Hornik, K., Stinchcombe, M., White, H. and Auer, P. (1994). “Degree of approximation results for feedforward networks approximating unknown mappings and their derivatives”, Journal of Neural Computation, 6, 1262–1275.
• Prewitt, K. and Lohr, S. (2006). “Bandwidth selection in local polynomial regression using eigenvalues”, Journal of the Royal Statistical Society, series B, 68, 135–154.
• Ruppert, D., Sheather, M. P. and Wand, P. (1995). “An effective bandwidth selector for local least squares regression”, Journal of the American Statistical Association, 90, 1257–1270.
• Ruppert, D. and Wand, P. (1994). “Multivariate locally weighted least squares regression”, Annals of Statistics, 22, 1346–1370
• Stone, C. J. (1980). “Optimal Rates of Convergence for Nonparametric Estimators”, The Annals of Statistics, 8, 1348–1360.
• Wang, K. and Gasser, T. (1996). “Optimal Rate for Estimating Local Bandwidth in Kernel Estimators of Regression Functions”, Scandinavian Journal of Statistics, 23, 303–312.