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 $X\prime$ and $Y\prime ,$ where $X\prime =X+\zeta$ and $Y\prime =Y+\eta .$ Our aim is to estimate nonparametrically $f_{X+Y}$ on the basis of random samples from the distributions of $X\prime ,\hspace{0.17em}Y\prime .$ 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.

Keywords:

• Consistency
• Convergence rate
• Density deconvolution
• Ordinary smooth and super smooth noises

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05
• 62G07
• 62G20

Acknowledgements

We would like to thank the anonymous reviewers and the editor for fruitful comments and suggestions which help to significantly improve the article. The first author (Le Thi Hong Thuy) would like to thank the support from Van Lang University, Vietnam for this study.