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W-transforms: Uniformity-preserving transformations and induced dependence structures

Marius Hofert, Zhiyuan Pang

TL;DR

This paper introduces W-transforms as uniformity-preserving, univariate transformations on the unit interval induced by a base distribution $F_X$ and a piecewise strictly monotone function $T$. It develops both continuous- and discontinuous-$F_X$ generalisations, derives the induced transformed copulas $C_{\bm{\mathcal{W}}}$, and provides analytical and structural results on their tail dependence, concordance, and symmetries. The framework enables copula-to-copula transformations, stochastic inverses, and ordinal-sum constructions, yielding flexible, tractable models that can capture asymmetry and tailored tail behavior. Through applications to tail removal, asymmetry creation, and the Danube dataset, the work demonstrates practical gains in dependence modelling and offers a versatile toolkit for constructing copulas with desired dependence features.

Abstract

W-transforms are introduced as uniformity-preserving univariate transformations on the unit interval induced by distribution functions and piecewise strictly monotone functions, and their properties are investigated. When applied componentwise to random vectors with standard uniform univariate margins, W-transforms naturally serve as copula-to-copula transformations. Properties of the resulting W-transformed copulas, including their analytical form, density, measures of concordance, tail dependence and symmetries, are derived. A flexible parametric family of W-transforms is proposed as a special case to further enhance tractability. Illustrative examples highlight the introduced concepts, and improved dependence modelling is demonstrated in terms of a real-life dataset.

W-transforms: Uniformity-preserving transformations and induced dependence structures

TL;DR

This paper introduces W-transforms as uniformity-preserving, univariate transformations on the unit interval induced by a base distribution and a piecewise strictly monotone function . It develops both continuous- and discontinuous- generalisations, derives the induced transformed copulas , and provides analytical and structural results on their tail dependence, concordance, and symmetries. The framework enables copula-to-copula transformations, stochastic inverses, and ordinal-sum constructions, yielding flexible, tractable models that can capture asymmetry and tailored tail behavior. Through applications to tail removal, asymmetry creation, and the Danube dataset, the work demonstrates practical gains in dependence modelling and offers a versatile toolkit for constructing copulas with desired dependence features.

Abstract

W-transforms are introduced as uniformity-preserving univariate transformations on the unit interval induced by distribution functions and piecewise strictly monotone functions, and their properties are investigated. When applied componentwise to random vectors with standard uniform univariate margins, W-transforms naturally serve as copula-to-copula transformations. Properties of the resulting W-transformed copulas, including their analytical form, density, measures of concordance, tail dependence and symmetries, are derived. A flexible parametric family of W-transforms is proposed as a special case to further enhance tractability. Illustrative examples highlight the introduced concepts, and improved dependence modelling is demonstrated in terms of a real-life dataset.

Paper Structure

This paper contains 23 sections, 17 theorems, 58 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The notion of a W-transform
  3. W-transforms constructed from continuous random variables
  4. Uniformity-preservation
  5. Properties of W-transforms
  6. A parametric family
  7. Generalised W-transforms
  8. W-transformed copulas
  9. Stochastic inverse of $\mathcal{W}$
  10. Componentwise W-transforms as multivariate measure-preserving transformations
  11. Tail dependence
  12. Concordance measures
  13. Symmetries
  14. Applications
  15. Removal of tail dependence in one tail
  16. ...and 8 more sections

Key Result

proposition 3.1

Let $X\sim F_X$ for continuous base distribution $F_X$, and let $T:D\to\mathbb{R}$ be pcsm with change points $(t_k)_{k=0}^K$, $K\in\bar{\mathbb{N}}$. If $U\sim\operatorname{U}(0,1)$, then $\mathcal{W}(U)\sim\operatorname{U}(0,1)$.

Figures (6)

  • Figure 1: 659 pseudo-observations of the Danube dataset of belzile2023 (left) and generated sample of the same size of a model constructed based on W-transforms introduced later (right).
  • Figure 10: A sample from the radially symmetric Cauchy copula with Spearman's rho $\rho_{\text{S}}=0$ (left), and a sample of the same size generated from the W-transformed Cauchy copula with $\rho_{\text{S}} \approx 0.47$ (right).
  • Figure 11: W-transform \ref{['eq:W:id:piece']} (left), a sample of size 2000 from the $t$-copula $C_{\nu=2,\rho=0.9}$ (centre) and a sample of the same size from the corresponding homogeneous W-transformed copula $C_{\mathcal{W}}$ (right).
  • Figure 12: W-transform $\mathcal{W}_{0.45}$\ref{['eq:W:asym']} (left), a sample of size 2000 from the $t$-copula $C_{\nu=2, \rho=0.9}$ (centre) and a sample of the same size from the corresponding W-transformed copula $C_{(\mathcal{W}_{0.3}, \mathcal{W}_{0.45})}$ (right).
  • Figure 13: Simulated sample of size 659 of the fitted Gumbel (left) and of the fitted Khoudraji-transformed Gumbel copula (right).
  • ...and 1 more figures

Theorems & Definitions (48)

  • definition 2.1: W-transforms
  • remark 2.2: Technical details
  • example 2.3: Non-uniformity-preservation of generic W-transforms
  • proposition 3.1: Uniformity-preservation under continuous $F_X$
  • example 3.2: V-transforms and their use in mcneil2021
  • proposition 3.3: Properties of W-transforms
  • proposition 3.4: Composition of W-transforms preserves properties of W-transforms
  • example 3.5: W-transforms
  • proposition 3.6: Sufficient conditions for $\mathcal{W}$ to be piecewise linear
  • lemma 3.7: Characterisation of uniformity-preservation under differentiability
  • ...and 38 more