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A general many quantiles + noise model is studied in the robust formulation(allowing non-normal, non-independent observations), where the identifiabilityrequirement for the noise is formulated in terms of quantiles rather than thetraditional zero expectation assumption. We propose a penalization method basedon the quantile loss function with appropriately chosen penalty function makinginference on possibly sparse high-dimensional quantile vector. We apply a localapproach to address the optimality by comparing procedures to the oraclesparsity structure. We establish that the proposed procedure mimics the oraclein the problems of estimation and uncertainty quantification (under the socalled EBR condition). Adaptive minimax results over sparsity scale follow fromour local results.

Robust oracle estimation and uncertainty quantification for possibly sparse quantiles