Publication:
Distribution Estimation of a Sum Random Variable from Noisy Samples
Distribution Estimation of a Sum Random Variable from Noisy Samples
No Thumbnail Available
Files
Date
2021
Authors
Cao Xuan Phuong
Le Thi Hong Thuy
Journal Title
Journal ISSN
Volume Title
Publisher
Research Projects
Organizational Units
Journal Issue
Abstract
Let X, Y be independent continuous univariate random variables with unknown distributions. Suppose we observe two independent random samples X′1,…,X′n and Y′1,…,Y′m from the distributions of X′=X+ζ and Y′=Y+η, respectively. Here ζ, η are random noises and have known distributions. This paper is devoted to an estimation for unknown cumulative distribution function (cdf) FX+Y of the sum X+Y on the basis of the samples. We suggest a nonparametric estimator of FX+Y and demonstrate its consistency with respect to the root mean squared error. Some upper and minimax lower bounds on convergence rate are derived when the cdf’s of X, Y belong to Sobolev classes and when the noises are Fourier-oscillating, supersmooth and ordinary smooth, respectively. Particularly, if the cdf’s of X, Y have the same smoothness degrees and n=m, our estimator is minimax optimal in order when the noises are Fourier-oscillating as well as supersmooth. A numerical example is also given to illustrate our method.