Abstract

The deconvolution problem in a classical measurement error model was investigated where contaminated observations were given by Y_j=X_j+\varepsilon_j,\ \ \varepsilon_j\simN(0,\sigma^2\ ),\ with unknown noise variance \sigma^2. The objective was to recover the cumulative distribution function F_X of the latent target variable X from a single sample of noisy observations. In contrast to the majority of existing literature which primarily focused on density estimation, a direct inference on F_X\ , a functionally rich object with broad applications in risk assessment, hypothesis testing was aimed and classified. A twostep semiparametric estimation procedure was proposed. First, the noise variance \sigma^2 was directly estimated from the contaminated data. Second, the resulting estimator was plugged into a deconvolution object with broad applications in risk assessment, hypothesis testing was testetd. A twobased reconstruction method to obtain an estimator of F_X. when F_X belonged to an ordinary smooth class, characterized by the polynomial decay of the characteristic function within a Sobolevtype regularity framework, namely |\varphi_X\ (t)|\asymp(1+|t|)^(-\alpha), the proposed estimator attaining the convergence rate (\ln\funcapplyn\ )^(-(\alpha+1\/2)\ ) was established. This rate coincided with the minimaxoptimal rate achieved in the case of known noise variance, demonstrating a full adaptivity with respect to the unknown noise level without any loss of statistical efficiency. Importantly, this approach did not require any auxiliary sample, such as pure noise observations or repeated measurements, making it particularly suitable for experimental settings with limited data availability. To the best of our knowledge, this was the first work providing a complete theoretical guarantee for distribution function estimation in the Gaussian deconvolution model with unknown variance in the ordinary smooth regime.