Deconvolution of ℙ(X<Y) with unknown error distributions

Abstract

This paper is devoted to a nonparametric estimation of the probability $\theta :=\mathbb{P}(X<Y),$ where X, Y are continuous univariate random variables of interest and observed with additional random errors. We focus on the case where the distributions of the random errors are unknown but symmetric around zero and can be estimated from some additional samples. Using deconvolution techniques, we propose an estimator of θ which depends on a regularization parameter. We then establish upper and lower bounds on convergence rate of the estimator under mean squared error when error densities are assumed to be supersmooth.

Keywords:

• Deconvolution
• characteristic function
• supersmooth errors

MSC 2010:

• 62G05
• 62G20

Acknowledgements

We would like to thank the reviewers for fruitful comments and suggestions which help to significantly improve the paper.