New perspectives on knockoffs construction

Let Λ be the collection of all probability distributions for (X, X̃), where X is a fixed random vector and X̃ ranges over all possible knockoff copies of X (in the sense of [4]). Three topics are developed in this note: (i) A new characterization of Λ is proved; (ii) A certain subclass of Λ, defined in terms of copulas, is introduced; (iii) The (meaningful) special case where the components of X are conditionally independent is treated in depth. In real problems, after observing X = x, each of points (i)-(ii)-(iii) may be useful to generate a value x̃ for X̃ conditionally on X = x. MSC 2010 subject classifications: Primary 62E10, 62H05; secondary 60E05, 62J02.