Density estimation of a sum random variable from contaminated data samples

Abstract

Let X, Y be independent random variables with unknown distributions and f X + Y be the unknown density of X + Y. Under the effects of independent random noises ζ and η, which are assumed to have known distributions, we observe the random variables X ′ and Y ′ , where X ′ = X + ζ and Y ′ = Y + η . Our aim is to estimate nonparametrically f X + Y on the basis of random samples from the distributions of X ′ , Y ′ . Using the observed data, we suggest an estimator of f X + Y and show that it is consistent with respect to the mean integrated squared error. Under some conditions restricted to the smoothness of the noises as well as of the variable X + Y, we derive some upper and lower bounds on the convergence rate of the error. We also conduct some simulations to illustrate the efficient of our method.