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 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.