The convolution theorem for estimating linear functionals in indirect nonparametric regression models

Nonparametric regression—directly or indirectly observed—is one of the important statistical models. On one hand it contains two infinite dimensional parameters (the regression function and the error density), and on the other it is of rather simple structure. Therefore, it may serve as an interesting paradigm for illustrating or developing abstract statistical theory for non-Euclidean parameters. In this paper estimation of a linear functional of the indirectly observed regression function is considered, when a deterministic design is used. It should be noted that any Fourier coefficient of an expansion of the regression function in an orthonormal basis is such a functional. Because the design is deterministic the observables are independent but not identically distributed. Local asymptotic normality is established and applied to prove Hájek's convolution theorem for this functional. Pertinent references are Beran [1977. Robust location estimates. Ann. Statist. 5, 431–444] and McNeney and Wellner [2000. Application of convolution theorems in semiparametric models with non-i.i.d. data. J. Statist. Plann. Inference 91, 441–480]. For purposes explained above, however, the paper is kept self-contained and full proofs are provided.