🤖 AI Summary
This study addresses the estimation of the distribution function of a latent variable in an additive measurement error model, allowing the latent distribution to be arbitrary—discrete, continuous, or mixed—without requiring the existence of a density or global smoothness assumptions. Building upon Fourier inversion and the algebraic structure of the estimator introduced by Mynbaev et al., the authors develop a direct estimation approach that, for the first time, accommodates generalized distributions with multiple jump points. Theoretical analysis provides non-asymptotic bounds on bias and variance and establishes the asymptotic unbiasedness and consistency of the proposed estimator. Simulation studies demonstrate superior finite-sample performance compared to existing methods and confirm the practical feasibility of the recommended parameter selection scheme.
📝 Abstract
We consider the classical additive measurement-error model $X=Y+Z$, where the latent random variable $Y$ has unknown distribution $F_Y$ and the error $Z$ has a known distribution. We develop direct estimators for three functionals of $F_Y$: (i) $F_Y(x)$ at continuity points; (ii) interval probabilities $F_Y(y)-F_Y(x)$ when $x<y$ are continuity points; and (iii) the size of a jump at a prespecified discontinuity. We derive non-asymptotic bias and variance bounds, and establish asymptotic unbiasedness and consistency. Unlike previous work, we do not require $F_Y$ to admit a density, have a mixture representation, or satisfy global Sobolev smoothness assumptions. The framework accommodates arbitrary latent distributions, including those with both discrete and continuous components, and distributions with multiple jumps. These results rely on a link between Fourier inversion theorems and the algebraic structure of a class of estimators proposed in Mynbaev, Martins-Filho and Henderson (2022). A simulation study evaluates feasible tuning procedures and, where available, compares the finite-sample performance of the proposed estimators with existing methods.