Parametric and nonparametric estimators in Fourier series semiparametric regression and their characteristics

Abstract

Consider data pairs 1 1 ( ,..., , ,..., , ) i ir i ip i x x t t y involving in a semiparametric regression model ( , ) i i i i y x t      , where 1 ( , ) ( ), p i i i j ji j x t x g t        1, , ; i n  1,..., j p  is the semiparametric regression curve. Response variable i y is assumed to be proportional to predictor variable 1 ( ,..., ) i i ir x x x   , but at the same time, its relationship with other predictor variables 1 ( ,..., ) i i ip t t t  is unidentified. The i x   and ( ) j ji g t are, parametric and nonparametric components respectively. In this study, the nonparametric component is approximated by Fourier series which is expressed by