Acta Applicandae Mathematicae ( IF 0.974 ) Pub Date : 2020-06-29 , DOI: 10.1007/s10440-020-00343-9
Cao Xuan Phuong

Let $$Y$$, $$X$$ and $$\varepsilon$$ be continuous univariate random variables satisfying the model $$Y = X + \varepsilon$$. Herein $$X$$ is of interest, $$Y$$ is a noisy version of $$X$$, and $$\varepsilon$$ is a random noise independent of $$X$$. This paper is devoted to a nonparametric estimation of cumulative distribution function $$F_{X}$$ of $$X$$ on the basis of independent random samples $$(Y_{1}, \ldots , Y_{n})$$ and $$(\varepsilon '_{1}, \ldots , \varepsilon '_{m})$$ drawn from the distributions of $$Y$$ and $$\varepsilon$$, respectively. We provide an estimator for $$F_{X}$$ based on a direct inversion formula and the ridge-parameter regularization. Our estimator is shown to be mean consistency with respect to the mean squared error whenever the set of all zeros of the characteristic function of $$\varepsilon$$ has Lebesgue measure zero. We then derive some convergence rates of the mean squared error uniformly on a nonparametric class for $$F_{X}$$ and on some different regular classes for the density of $$\varepsilon$$. A numerical example is performed to illustrate the efficiency of our method.

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