Distribution Estimation of a Sum Random Variable from Noisy Samples

Abstract

Let X, Y be independent continuous univariate random variables with unknown distributions. Suppose we observe two independent random samples $$X'_1, \ldots , X'_n$$ and $$Y'_1, \ldots , Y'_m$$ from the distributions of $$X' = X+\zeta $$ and $$Y'=Y+\eta $$ , respectively. Here $$\zeta $$ , $$\eta $$ 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.