Distribution Estimation of a Sum Random Variable from Noisy Samples

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 dist...

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Bibliographic Details
Published in:Bulletin of the Malaysian Mathematical Sciences Society 2021, Vol.44 (5), p.2773-2811
Main Authors: Phuong, Cao Xuan, Thuy, Le Thi Hong
Format: Article
Language:English
Subjects:
Applications of Mathematics
Continuity (mathematics)
Distribution functions
Independent variables
Lower bounds
Mathematics
Mathematics and Statistics
Minimax technique
Random variables
Smoothness
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Summary: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) F X + Y of the sum X + Y on the basis of the samples. We suggest a nonparametric estimator of F X + 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.
ISSN:0126-6705
2180-4206
DOI:10.1007/s40840-021-01088-w