Asymptotics of a wavelet estimator in the nonparametric regression model with repeated measurements under a NA error process

Abstract

Consider the nonparametric regression model with repeated measurements: \(Y^{(j)}(x_{ni})=g(x_{ni})+e^{(j)}(x_{ni})\), where \(Y^{(j)}(x_{ni})\) is the \(j\)th response at the point \(x_{ni}\), \(x_{ni}\)’s are known and nonrandom, and \(g(\cdot )\) is unknown function defined on a closed interval \([0,1]\). For exhibiting the correlation among the units and avoiding any assumptions among the observations within the same unit, we consider the model with negative associated (NA) error structures, that is, \(\{e^{(j)}(x), j\ge 1\}\) is a mean zero NA error process. The wavelet procedures are developed to estimate the regression function. Some asymptotics of wavelet estimator are established under suitable conditions.