Acta Applicandae Mathematicae ( IF 1.6 ) 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.
中文翻译:
具有未知噪声分布的累积分布函数的反卷积
令\(Y \),\(X \)和\(\ varepsilon \)是满足模型\(Y = X + \ varepsilon \)的连续单变量随机变量。这里\(X \)是令人感兴趣的,\(Y \)是\(X \)的嘈杂版本,\(\ varepsilon \)是独立于\(X \)的随机噪声。本文专门累积分布函数的非参数估计\(F_ {X} \)的\(X \)独立随机样本的基础上\((Y_ {1},\ ldots,Y_ {N})\ )和\((\ varepsilon'_ {1},\ ldots,\ varepsilon'_ {m})\)分别从\(Y \)和\(\ varepsilon \)的分布得出。我们基于直接反演公式和脊参数正则化为\(F_ {X} \)提供了一个估算器。每当\(\ varepsilon \)的特征函数的所有零的集合都具有Lebesgue度量为零时,我们的估计量就相对于均方误差而言表现为均值一致性。然后,我们针对\(F_ {X} \)的非参数类和针对\(\ varepsilon \)的密度的某些不同常规类,均匀地得出均方误差的收敛速度。数值例子说明了我们方法的有效性。