Calibrated Order-Randomized Rosenblatt Tests

2026-09-27

AI Summary

This study addresses the instability of statistical power in Rosenblatt-transform-based goodness-of-fit tests for multivariate distributions, which arises from their dependence on coordinate ordering. To overcome this limitation, we propose a calibrated uniformity test based on randomized coordinate permutations and evidence combination. Theoretically, we demonstrate that reordering merely redistributes the energy of the Mahalanobis signal. Accordingly, we design a calibration strategy integrating two-sided statistics with a re-estimated parametric bootstrap to restore nominal significance levels. Simulation experiments and an application to foreign exchange risk modeling indicate that the proposed method substantially enhances testing power and effectively identifies market anomalies associated with events such as Brexit and the COVID-19 pandemic.

Abstract

We test whether a multivariate vector X conforms to a specified distribution F, a problem in copula modelling and density forecasting. The Rosenblatt transform reduces it to a test of uniformity, but depends on an arbitrary coordinate ordering that strongly affects power under dependence. We study order randomization: applying the transform under many random orderings and merging the evidence with dependence-robust rules. Reordering conserves the total Mahalanobis signal energy and merely redistributes it, so one ordering is a lucky or unlucky draw. In simulations we observe significant gains in calibrated power over both the expected single random ordering and order-invariant references. Two ingredients are essential: a two-sided base statistic, and a re-estimating parametric bootstrap that restores level under an estimated null and unlocks the gain. The calibrated pooled tests are robust to the departure's shape; no order-invariant reference we compare is: the symmetric-root test collapses on diffuse departures, while the shape-flat chi-squared test trails on concentrated ones. We apply it to a Gaussian foreign-exchange risk model over a decade of daily data on nine currencies, where it detects episodes such as Brexit and COVID. Though we focus on Gaussian nulls, the procedure extends to any null whose conditional distributions can be computed and simulated from.

Problem

• Goodness-of-fit testing
• Rosenblatt transform
• Copula modelling
• Density forecasting
• Order randomization

Innovation

• Order-Randomized Rosenblatt Transform
• Calibrated Power
• Re-estimating Parametric Bootstrap
• Two-sided Base Statistic
• Copula Modelling