Robust estimation in inverse problems via quantile coupling

In this article we consider a sequence of hierarchical space model of inverse problems. The underlying function is estimated from indirect observations over a variety of error distributions including those that are heavy-tailed and may not even possess variances or means. The main contribution of this paper is that we establish some oracle inequalities for the inverse problems by using quantile coupling technique that gives a tight bound for the quantile coupling between an arbitrary sample p-quantile and a normal variable, and an automatic selection principle for the nonrandom filters. This leads to the data-driven choice of weights. We also give an algorithm for its implementation. The quantile coupling inequality developed in this paper is of independent interest, because it includes the median coupling inequality in literature as a special case.