Two stage sequential procedure for nonparametric regression estimation

Abstract

In nonparametric statistics the functional form of the relationship between the response variable and its associated predictor variables is unspecified but it is assumed to be a smooth function. We develop a procedure for constructing a fixed width confidence interval for the predicted value at a specified point of the independent variable. The optimal sample size for constructing this interval is obtained using a two stage sequential procedure which relies on some asymptotic properties of the Nadaraya--Watson and local linear estimators. Finally, a large scale simulation study demonstrates the applicability of the developed procedure for small and moderate sample sizes. The procedure developed here should find wide applicability since many practical problems which arise in industry involve estimating an unknown function. References E. F. Schuster. Joint asymptotic distribution of the estimated regression function at a finite number of distinct points. Ann. Math. Statist., 43, 1972, 84--88. http://www.jstor.org/stable/2239900 E. Isogai. The convergence rate of fixed-width sequential confidence intervals for a probability density function. Seq. Anal., 6, 1987, 55--69. doi:10.1081/SQA-100102647 E. A. Nadaraya. On estimating regression. Theory of Probab. Appl., 9, 1964, 141--142. doi:10.1137/1109020 M. P. Wand and M. C. Jones. Kernel Smoothing. Chapman and Hall, London, 1995. G. S. Watson. Smooth regression analysis. Sankhya Series A, 26, 1964, 359--372. W. S. Cleveland. Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc., 74, 1979, 829--836. http://www.jstor.org/stable/2286407 J. Fan and I. Gijbels. Local Polynomial Modelling and its Applications. Chapman and Hall, London, 1996. T. Gasser, L. Sroka and C. Jennen-Steinmetz. Residual variance and residual pattern in nonlinear regression. Biometrika, 73, 1986, 625--633. http://www.jstor.org/stable/2336527 M. Ghosh, N. Mukhopadhyay and P. K. Sen. Sequential Estimation. Wiley, New York, 1997.