Acta Mathematica Vietnamica
The Cumulative Distribution Function of a Mixture Model with Normal Error
Dang Duc Trong
,
Nguyen Hoang Thanh
,
Nguyen Tien Dat
,
Thai Phuc Hung
Abstract
This paper studies statistical deconvolution, focusing on estimating the cumulative distribution function $F_X$ of a random variable X from observations $Y_j = X_j + \sigma \xi _j$, where $X_j$ and $\xi _j$ are independent copies of $X$ and $\xi$, respectively. We assume that the noise $\xi _j$ is normally distributed with zero mean and unknown variance $\sigma ^2$. Under this framework, we propose a new semi-parametric estimator for $F_X$. Our results show that, over a subclass of Sobolev-type distributions, the estimator achieves a convergence rate comparable to previous studies, demonstrating that the accurate semi-parametric inference for $F_X$ is feasible even when the noise variance is unknown.
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