Order invariance of the generalized Rosenblatt transform on standard Borel spaces

Abstract

The Rosenblatt transform maps a random vector to independent uniform variables through successive conditional distribution functions; for laws on $\mathbb{R}^{\mathbb{N}}$ with atomless conditional distributions and a fixed order of the coordinates, it is the almost surely unique monotone triangular map with this property. For a fixed reference law, transform-based tests remain valid under every fixed admissible order, but that choice can still affect their statistical power. We extend the generalized Rosenblatt transform to laws on countable products of standard Borel spaces, which requires, in addition to an order on the index set, a total order on each component space. For fixed orders, we establish identification and uniqueness results for this extended generalized Rosenblatt transform (EGRT) that parallel those for the classical transform. We then characterize order dependence in two senses: first demonstrating conditions under which the EGRT is invariant under linear extensions of prescribed partial orders, and second showing that under every admissible order the output of the EGRT retains all the statistical information from the input observations over reference-dominated families. We close with a brief discussion of what these results mean for the relationship between order choice and goodness-of-fit testing.