Deconvolution of Cumulative Distribution Function with Unknown Noise Distribution

• Published in: Acta applicandae mathematicae Vol. 170; no. 1; pp. 483 - 514
• Main Author: Phuong, Cao Xuan
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Characteristic functions; Computational Mathematics and Numerical Analysis; Continuity (mathematics); Distribution functions; Mathematics; Mathematics and Statistics; Partial Differential Equations; Probability Theory and Stochastic Processes; Random noise; Random variables; Regularization; Convergence rate; Non-standard noise; 62G20; 62G07; Mean consistency; Cumulative distribution function
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-020-00343-9

Let Y , X and ε be continuous univariate random variables satisfying the model Y = X + ε . Herein X is of interest, Y is a noisy version of X , and ε 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 , … , Y n ) and ( ε 1 ′ , … , ε m ′ ) drawn from the distributions of Y and ε , 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 ε 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 ε . A numerical example is performed to illustrate the efficiency of our method.