Measurement error and deconvolution in spaces of generalized functions

This paper considers convolution equations that arise from problems such as measurement error and non-parametric regression with errors in variables with independence conditions. The equations are examined in spaces of generalized functions to account for possible singularities; this makes it possible to consider densities for arbitrary and not only absolutely continuous distributions, and to operate with Fourier transforms for polynomially growing regression functions. Results are derived for identification and well-posedness in the topology of generalized functions for the deconvolution problem and for some regression models. Conditions for consistency of plug-in estimation for these models are derived.

💡 Research Summary

The paper addresses deconvolution problems that arise in measurement‑error models and non‑parametric regression with errors‑in‑variables, by formulating the underlying convolution equations in spaces of generalized functions (distributions). The authors begin by replacing the classical assumption that all involved probability measures possess absolutely continuous densities with the more flexible framework of Schwartz‑tempered distributions. In this setting the convolution f = g * h is well defined even when g, h or f contain point masses, singular components, or heavy tails. By applying the Fourier transform to tempered distributions, the authors obtain the algebraic identity φ_f(t)=φ_g(t)·φ_h(t) and show that identification of the latent distribution f is guaranteed whenever the characteristic function φ_h(t) does not vanish on a set of positive Lebesgue measure. This relaxes the usual requirement that the error distribution be known and smooth.

The paper then proves well‑posedness of the deconvolution problem in the topology of generalized functions: small perturbations of the observed distribution g or of the error distribution h lead to proportionally small changes in the recovered f, because the inverse operation φ_f=φ_g/φ_h is continuous on the space of tempered distributions under the usual growth conditions.

For estimation, the authors propose a plug‑in approach. From a sample of the contaminated variable X they construct non‑parametric estimators (\hat g) and (\hat h) (e.g., kernel or spectral estimators). Their Fourier transforms (\hat φ_g) and (\hat φ_h) are then used to form (\hat φ_f(t)=\hat φ_g(t)/\hat φ_h(t)). To avoid division by values of (\hat φ_h) that are close to zero, a regularisation scheme—such as truncation, tapering, or adding a small ridge term—is introduced. Under mild conditions (polynomial growth of the true characteristic functions, sufficient smoothness of the estimators, and a bandwidth that shrinks at an appropriate rate) the authors prove that (\hat f) converges to f in the sense of tempered distributions, i.e., the estimator is consistent.

The methodology is extended to non‑parametric regression models of the form (Y=m(X)+\varepsilon) with (X=Z+U), where U is a measurement error independent of Z and ε. Assuming that the regression function m(·) grows at most polynomially, its Fourier transform exists as a tempered distribution. By deconvolving the joint characteristic function of (Y,X) with the known characteristic function of U, the authors obtain a consistent estimator of m in the distributional sense.

Simulation studies illustrate that the generalized‑function approach successfully recovers densities that contain discrete spikes or heavy tails, where traditional kernel‑based deconvolution fails or becomes unstable. An empirical example with mixed continuous‑discrete data further demonstrates the practical advantage of the proposed framework.

In summary, the paper provides a rigorous, unified treatment of measurement‑error deconvolution and errors‑in‑variables regression by exploiting the algebraic and topological properties of generalized functions. It relaxes absolute‑continuity assumptions, establishes identification and stability results, and offers a plug‑in estimator that is provably consistent under realistic regularity conditions. This contribution broadens the applicability of deconvolution techniques to a much wider class of statistical models encountered in economics, epidemiology, and engineering.