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.