Asymptotically efficient estimation based on wavelet of expectation value in a partial linear model

This paper is concerned with a semiparametric regression model Y_i = θ₁+θ₂(T_i)+ε_i, i = 1, ⋯, n where the error ε_i is independent of T_i for each i, θ₁ is an unknown constant of interest, and θ₂ is an unknown function on [0, 1]. In order to obtain an asymptotically efficient estimate θ̂_1wof θ₁, θ₂(t) is generally in the literature assumed to be a continuously differentiable function on [0, 1]. In this paper, it is constructed assuming only that θ₂ is an unknown piecewise differentiable function on [0, 1]. In the process of construction, wavelet expansion is employed.