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.

Non-Parametric Sequential Estimation of a Regression Function Based on Dependent Observations

“…There are a lot of results and publications dedicate this problem for independent and dependent observations, in asymptotic and non-asymptotic problem statements (see, e.g, Bai and Liu (2007), Dharmasena at al. (2008)-Hondaa (1998, Leonov (1999), Nazin and Ljung (2002), )-Scott (1992 etc.)…”

“…(2008)-Hondaa (1998, Leonov (1999), Nazin and Ljung (2002), )-Scott (1992 etc.) It should be noted, that often regressors X a supposed to be non-random or bounded and noises ∆ of the model are independent, see Bai and Liu (2007); Dharmasena at al. (2008); Györfi at al.…”

“…(2012); Konev (1985); Küchler and Vasiliev (2010) among others). Sequential approach has also been applied to non-parametric regression and density function estimation problems as well (see, e.g., Carroll (1976); Davies and Wegman (1975), Dharmasena at al. (2008)-Efroimovich (2007, Hondaa (1998); Liski at al.…”