An arXiv preprint examines what happens when monotone rearrangement, a method for reshaping an estimate into a monotone pattern, is used while the underlying target has flat regions. The analysis covers density estimation and copula settings, giving separate asymptotic results for each.
The setups are theoretical and ask what happens in large samples under stated probability models. The density component is a histogram analysis based on an i.i.d. uniform sample; the copula component assumes independent X and Y with continuous marginal distributions.
The histogram result breaks at the edges
For the uniform-density model, the rearranged histogram settles into a limiting random curve after deterministic centering, meaning a fixed adjustment is made before its fluctuations are studied. This process convergence holds on compact interior intervals, away from the ends, at the parametric rate. The rate is independent of the number of histogram cells if the required growth conditions hold.
At the boundary, the result changes. Process convergence cannot be extended to the boundary; instead, the supremum, or largest value in the process, has a separate Gumbel limit, an extreme-value limit. That result has its own scaling and growth conditions.
The interior limit is Gaussian, built from a Brownian-bridge-based process and a jointly normal centering variable. The proof uses a Hungarian construction to couple the empirical process with standard Brownian bridges, providing the Gaussian approximation behind the result.
Copula estimates converge across the domain
Copulas, which describe dependence separately from variables' individual distributions, form the second setting. The study examines two rearranged constructions: one based on empirical increments and one based on a checkerboard derivative.
Under independence, both centered and rescaled rearranged copula estimators converge weakly at the parametric rate over the full square to an integrated centered Gaussian process. Put simply, the entire estimated surface has one Gaussian large-sample description across its domain. The authors link the full-square result to the integrating step.
With suitable bandwidth conditions, the checkerboard estimator is asymptotically equivalent to the empirical-increment estimator. This describes their large-sample behavior; it is not a finite-sample performance ranking.
The limiting copula process is specified through a standard Brownian bridge, a centered Gaussian process and an explicit covariance structure. But it remains model-conditional: the weak-convergence result is derived under independence, not for arbitrary dependent copulas.
A route to dependence measures
To move from estimated processes to numerical dependence summaries, the paper applies a functional delta method, which transfers a process limit through an appropriate derivative. It gives asymptotic distributions for rearranged dependence-measure estimators; when the relevant derivative is linear and continuous, the limits are normal.
Spearman's rho is used as the worked example of this framework.
Where the conclusions apply
The document is an arXiv preprint, and its main conclusions are asymptotic statements under the stated models, bandwidth conditions and cell-growth conditions.
That scope sets clear boundaries: histogram process convergence stays on interior intervals; the copula result assumes independence; and normality for dependence measures requires linear, continuous derivatives. The study therefore does not offer a general result for arbitrary dependence or boundary behavior.
Paper data and sources
Original title: Limiting properties of monotone rearrangements of estimators when the truth is flat
Authors: Holger Dette, Marius Kroll, Stanislav Volgushev
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text