Quantile-Based Reparametrisation

• Quantile-based reparametrisation is a technique that represents a target distribution as the image of a simpler reference law under a quantile or transport map.
• It extends the classic inverse-CDF method to complex settings such as Bayesian posterior simulations, manifold-valued data, and regression models.
• Numerical strategies, including reconditioning and branch selection, are essential to mitigate tail instability and ensure high-quality sampling.

Searching arXiv for the papers on arXiv and closely related work to ground the article in current literature. Quantile-based reparametrisation denotes the representation of a random variable, random vector, process, or transformed sample as the image of a simpler reference input under a quantile map or transport map. In the univariate case, if $F$ is a cumulative distribution function and $Q(u)\equiv F^{-1}(u)$, the construction is the inversion formula $X=Q(U(0,1))$ (Pedersen, 2017). In the broader literature, the same idea appears as a conditional vector quantile map for Bayesian posteriors, a manifold-valued optimal transport map $Y=Q(U)$, a parametric quantile family $Q(u;\theta)$ used for inference, and a rank-based transformation $\Phi_f(x)=\Pi_x f$ used for supervised normalization (Kim et al., 2024, Hallin et al., 2024, Chattopadhyay et al., 28 Nov 2025, Morvan et al., 2017). The common structure is a reference law together with a monotone or transport-based map that specifies the target distribution directly through quantiles rather than through a likelihood, density, or moment system.

1. Canonical formulation in one dimension

The basic univariate formalism is inverse-CDF sampling. If $F$ is the CDF of a density $f$, the quantile function is $Q(u)\equiv F^{-1}(u)$, and inversion-based sampling draws $X=Q(U(0,1))$ (Pedersen, 2017). In this sense, quantile-based reparametrisation is exact in the reals: the distribution is generated by pushing a uniform variate through the quantile map.

This formulation recurs in parametric settings in which the quantile function itself is taken as primitive. A quantile-parametrized family is specified by a known quantile function $Q(u)\equiv F^{-1}(u)$0 for $Q(u)\equiv F^{-1}(u)$1, with the inverse relationships $Q(u)\equiv F^{-1}(u)$2 and $Q(u)\equiv F^{-1}(u)$3 on the support (Chattopadhyay et al., 28 Nov 2025). This is especially important when the quantile function is available in closed form but the CDF or density is not. The literature on Tukey Lambda and generalized Lambda families adopts exactly this viewpoint, treating $Q(u)\equiv F^{-1}(u)$4 as the defining object of the model rather than as a derived function (Chattopadhyay et al., 28 Nov 2025).

A related parametric use appears in distributional regression. Quantile index regression postulates that, on a quantile interval $Q(u)\equiv F^{-1}(u)$5, the conditional quantile function can be written as

$Q(u)\equiv F^{-1}(u)$6

where the low-dimensional quantile indices $Q(u)\equiv F^{-1}(u)$7 are linked to covariates through componentwise link functions (Zhang et al., 2021). This is explicitly described as a quantile-based reparametrisation of the conditional distribution because the full conditional law is specified via its conditional quantile function.

The same principle also underlies distributional reparametrisation in positive asymmetric families. In the Singh–Maddala and Dagum models, the scale parameter is replaced by a chosen distributional quantile $Q(u)\equiv F^{-1}(u)$8, after which $Q(u)\equiv F^{-1}(u)$9 is linked to predictors in a regression equation (Saulo et al., 2022). Here the reparametrisation does not merely facilitate simulation; it changes the statistical parameterization so that inference and interpretation are centered on a policy-relevant quantile.

2. Numerical conditioning and robust inversion

Although $X=Q(U(0,1))$0 is exact in continuous mathematics, finite-precision arithmetic introduces two pathologies in the tails. First, for two-tailed distributions, $X=Q(U(0,1))$1 can be intrinsically ill-conditioned near $X=Q(U(0,1))$2: when $X=Q(U(0,1))$3 is steep, small perturbations in $X=Q(U(0,1))$4 produce large perturbations in $X=Q(U(0,1))$5 (Pedersen, 2017). Second, canonical floating-point uniforms on $X=Q(U(0,1))$6 have fixed absolute spacing $X=Q(U(0,1))$7, which forces ill-conditioning as $X=Q(U(0,1))$8 even when the mathematical condition number of $X=Q(U(0,1))$9 is acceptable there (Pedersen, 2017).

The sensitivity analysis follows from differentiating $Y=Q(U)$0, which gives $Y=Q(U)$1 (Pedersen, 2017). In the tails, where $Y=Q(U)$2 is small, $Y=Q(U)$3 is large. The paper “Reconditioning your quantile function” identifies the consequence: sparse floating-point coverage of extreme $Y=Q(U)$4-values causes repetitive samples, skipped representable floats, and loss of precision in the generated variates (Pedersen, 2017).

The proposed remedy couples the quantile map and the base variate. One component is “quantile flip-flop,” a two-quantile composition split at the median. In the exponential case, the two valid quantiles

$Y=Q(U)$5

are each used only on $Y=Q(U)$6, with random branch selection, so that both tails remain well-conditioned on their operating domains (Pedersen, 2017). The second component is a high-entropy uneven uniform variate on $Y=Q(U)$7, generated by Algorithm Upqr under IEEE-754 floating-point assumptions, with at least $Y=Q(U)$8 significant bits before rounding and an odd mantissa index to defeat ties-to-even bias (Pedersen, 2017).

The practical message is that quantile-based reparametrisation is a coupled numerical system rather than a purely symbolic identity. This directly contradicts the common assumption that a closed-form quantile function alone guarantees high-quality sampling. The cited analysis shows that cancellation, fixed-spacing uniforms, and inappropriate branch choices can destroy tail precision even when the quantile formula is mathematically correct (Pedersen, 2017).

3. Optimal-transport generalisations in multivariate and geometric settings

In multivariate Bayesian simulation, quantile-based reparametrisation is generalized through conditional optimal transport. “Deep Generative Quantile Bayes” learns a conditional vector quantile map $Y=Q(U)$9 that pushes forward a simple base law $Q(u;\theta)$0 on the unit ball $Q(u;\theta)$1 to an approximation of the posterior $Q(u;\theta)$2 (Kim et al., 2024). The learned generator is the Brenier map

$Q(u;\theta)$3

with $Q(u;\theta)$4 convex in $Q(u;\theta)$5. Sampling proceeds by drawing $Q(u;\theta)$6 and setting $Q(u;\theta)$7 (Kim et al., 2024). In one dimension this reduces to inverse transform sampling; in higher dimensions it is the Monge–Kantorovich vector quantile.

The conditional formulation is made tractable through the affine-in-summary parameterization

$Q(u;\theta)$8

where $Q(u;\theta)$9 and the components of $\Phi_f(x)=\Pi_x f$0 are parameterized by input convex neural networks and $\Phi_f(x)=\Pi_x f$1 is a learned summary statistic built from DeepSets for exchangeable data and LSTM components for dependent data (Kim et al., 2024). This yields a differentiable reparametrisation map and also enables direct sampling from Bayesian credible sets. Using MK depth and the identity

$\Phi_f(x)=\Pi_x f$2

the method samples a $\Phi_f(x)=\Pi_x f$3-credible set by restricting the latent draw $\Phi_f(x)=\Pi_x f$4 to the ball $\Phi_f(x)=\Pi_x f$5 before applying the generator (Kim et al., 2024). The paper states frequentist consistency for the learned quantile map, the recovered posterior in $\Phi_f(x)=\Pi_x f$6, and the associated credible sets in Hausdorff distance.

A geometric analogue is developed for compact boundaryless Riemannian manifolds. There the absence of a canonical order is replaced by an optimal transport map from a baseline measure $\Phi_f(x)=\Pi_x f$7, typically the uniform distribution induced by Riemannian volume, to the target law $\Phi_f(x)=\Pi_x f$8 (Hallin et al., 2024). The manifold quantile function $\Phi_f(x)=\Pi_x f$9 is the $F$0-a.s. unique optimal transport pushing $F$1 to $F$2, so that if $F$3, then $F$4 (Hallin et al., 2024). The inverse map $F$5 is the distribution function in the transport sense.

In this framework, quantile regions and contours are obtained by transporting reference geometric sets under $F$6. Empirical versions are constructed by discrete optimal transport between the sample and a structured grid on the manifold, yielding empirical ranks and signs that are distribution-free, with Glivenko–Cantelli-type uniform convergence for empirical distribution and quantile maps and uniform convergence of empirical contours in Hausdorff distance (Hallin et al., 2024). A plausible implication is that quantile-based reparametrisation provides, in manifolds as in Euclidean spaces, a replacement for total order by a transport-induced center-outward ordering.

4. Quantile-parametrized inference without closed-form likelihoods

A distinct inferential use of quantile-based reparametrisation arises when $F$7 is known but $F$8 or $F$9 is not. “Inference for quantile-parametrized families via CDF confidence bands” constructs confidence sets by inverting distribution-free CDF bands through the quantile map (Chattopadhyay et al., 28 Nov 2025). If $f$0 and $f$1 are lower and upper confidence bands for the empirical CDF, the parameter confidence set is

$f$2

The equivalent order-statistic form uses inequalities $f$3 for all $f$4 (Chattopadhyay et al., 28 Nov 2025).

The framework is band-agnostic, but the paper focuses on the Dvoretzky–Kiefer–Wolfowitz band and the Dümbgen–Wellner band. The latter is rate-optimal, variable-width, tighter near extremes, and requires Monte Carlo computation of its critical value under Uniform$f$5 (Chattopadhyay et al., 28 Nov 2025). The main guarantee is finite-sample coverage: if the band covers the true CDF values at all order statistics with probability at least $f$6, then the inverted parameter set also contains the true parameter with probability at least $f$7 (Chattopadhyay et al., 28 Nov 2025).

This approach is motivated by the failure of several conventional procedures in quantile families. The cited paper states that approximate maximum likelihood and related estimators can exhibit non-$f$8 and non-normal asymptotics in parts of the parameter space, and that bootstrap procedures can under-cover, especially for Tukey Lambda models near $f$9 and around $Q(u)\equiv F^{-1}(u)$0 (Chattopadhyay et al., 28 Nov 2025). Quantile-based inversion through CDF bands therefore functions as an assumption-lean alternative to likelihood-based inference.

A structurally related but model-specific reparametrisation appears in parametric income regression. In the quantile-regression formulations based on Singh–Maddala and Dagum families, the chosen $Q(u)\equiv F^{-1}(u)$1-quantile $Q(u)\equiv F^{-1}(u)$2 replaces the scale parameter, and $Q(u)\equiv F^{-1}(u)$3 is linked to covariates through a link function, usually the log link (Saulo et al., 2022). Maximum likelihood is then carried out in the reparametrized model. The paper reports asymptotic normality of the estimator with covariance given by the inverse expected Fisher information, and diagnostics based on generalized Cox–Snell and randomized quantile residuals (Saulo et al., 2022). This suggests that quantile-based reparametrisation can support both likelihood-free inversion and conventional likelihood inference, depending on whether the post-reparametrisation density remains tractable.

5. Regression, normalization, and process-level constructions

In regression, the principal appeal of quantile-based reparametrisation is that the conditional law is specified directly through a valid quantile family. Quantile index regression uses

$Q(u)\equiv F^{-1}(u)$4

on a target interval $Q(u)\equiv F^{-1}(u)$5, with tail behavior controlled by low-dimensional quantile indices such as Tukey or generalized Lambda parameters (Zhang et al., 2021). Estimation proceeds by composite quantile regression over a grid $Q(u)\equiv F^{-1}(u)$6, and the paper emphasizes that non-crossing is automatic because $Q(u)\equiv F^{-1}(u)$7 is non-decreasing in $Q(u)\equiv F^{-1}(u)$8 for admissible parameter values (Zhang et al., 2021). The theoretical results include consistency and asymptotic normality in low dimension, plus high-dimensional non-asymptotic $Q(u)\equiv F^{-1}(u)$9 and $X=Q(U(0,1))$0 error bounds under SCAD or MCP penalization (Zhang et al., 2021).

A preprocessing variant appears in supervised quantile normalization. Standard quantile normalization assigns target quantiles $X=Q(U(0,1))$1 to the sorted ranks of a sample $X=Q(U(0,1))$2, which can be written as

$X=Q(U(0,1))$3

where $X=Q(U(0,1))$4 is the permutation matrix induced by the sample ranks and $X=Q(U(0,1))$5 is the target quantile vector (Morvan et al., 2017). SUQUAN learns $X=Q(U(0,1))$6 jointly with the parameters of a downstream linear predictor by minimizing prediction loss subject to monotonicity, norm, and optionally smoothness constraints on $X=Q(U(0,1))$7 (Morvan et al., 2017). Because

$X=Q(U(0,1))$8

the problem is equivalent to low-rank matrix regression on permutation-matrix embeddings, with a rank-1 parameter matrix $X=Q(U(0,1))$9 (Morvan et al., 2017). This is a quantile-based reparametrisation of feature vectors rather than of a generative distribution.

Continuous-time constructions extend the same logic to stochastic processes. In quantile diffusions, a base diffusion $Q(u)\equiv F^{-1}(u)$00 is transformed through a composite distribution–quantile map. The random-level construction defines

$Q(u)\equiv F^{-1}(u)$01

so that the marginal law of $Q(u)\equiv F^{-1}(u)$02 is generated by applying a parametric quantile function $Q(u)\equiv F^{-1}(u)$03 to the uniformized diffusion $Q(u)\equiv F^{-1}(u)$04 (Brannelly et al., 2019). Applying Itô’s formula yields an explicit SDE for $Q(u)\equiv F^{-1}(u)$05, with drift and diffusion coefficients expressed through the base transition density and the target quantile density (Brannelly et al., 2019). Function-valued quantile diffusions instead let the parameter vector $Q(u)\equiv F^{-1}(u)$06 follow its own diffusion, so that the entire quantile function $Q(u)\equiv F^{-1}(u)$07 evolves over time (Brannelly et al., 2019). In both forms, quantile-based reparametrisation turns dynamic location, scale, skewness, and kurtosis into parameters of a quantile family rather than parameters of a density.

6. Approximation, guarantees, and recurring limitations

Several works make explicit that quantile-based reparametrisation is attractive precisely when likelihoods are unavailable, unstable, or inconvenient. In simulator-based Bayesian inference, the appeal is “no explicit likelihood needed,” together with differentiability of the generator and direct credible-set sampling (Kim et al., 2024). In quantile-parametrized families, the attraction is distribution-free inference without closed-form $Q(u)\equiv F^{-1}(u)$08 or $Q(u)\equiv F^{-1}(u)$09 and without relying on regular likelihood asymptotics (Chattopadhyay et al., 28 Nov 2025). In regression and normalization, the attraction is non-crossing structure, direct control of target quantiles, and task-coupled transformations (Zhang et al., 2021, Morvan et al., 2017).

At the same time, the literature is explicit about limitations. In multivariate transport models, valid quantile mappings require monotonicity or cyclical monotonicity; misspecified summaries $Q(u)\equiv F^{-1}(u)$10, very high-dimensional $Q(u)\equiv F^{-1}(u)$11, and insufficient training coverage can degrade transport quality (Kim et al., 2024). On manifolds, the framework depends on compactness, absolute continuity, cost regularity, cut-locus separation, and nonnegative cost-sectional curvature assumptions that can be difficult to verify beyond standard manifolds (Hallin et al., 2024). In quantile families, monotonicity in $Q(u)\equiv F^{-1}(u)$12 must hold, ties may require conservative treatment, and DKW bands can be conservative relative to Dümbgen–Wellner bands (Chattopadhyay et al., 28 Nov 2025). In numerical inversion, even a mathematically valid $Q(u)\equiv F^{-1}(u)$13 can lose dozens of ULPs in the tails if the base uniform and the internal arithmetic are poorly chosen (Pedersen, 2017).

Analytical approximation of quantile maps is itself an active topic. A Weibull-based polynomial transformation model approximates a target quantile by

$Q(u)\equiv F^{-1}(u)$14

with coefficients estimated either by percentile matching against known target quantiles or by probability-weighted moments matching for empirical data (Xiao, 2015). The paper reports accurate approximations within the probit range $Q(u)\equiv F^{-1}(u)$15 and notes that the resulting map directly supports inverse-CDF sampling and reparameterization gradients (Xiao, 2015). This suggests that even when a quantile map is unavailable in closed form, the quantile-based viewpoint can still be retained through an explicit approximation.

A recurring misconception is that quantile-based reparametrisation is only inverse-CDF sampling for scalar variables. The cited literature shows a broader picture: conditional Brenier maps for posteriors, c-convex transport maps on manifolds, inversion of empirical CDF bands for finite-sample confidence sets, rank-based low-rank normalization maps, and continuous-time quantile diffusions all instantiate the same principle under different structural assumptions (Kim et al., 2024, Hallin et al., 2024, Chattopadhyay et al., 28 Nov 2025, Morvan et al., 2017, Brannelly et al., 2019). Another misconception is that quantile parameterization removes the need for careful computation; the numerical and theoretical results instead show that stability, monotonicity, and regularity conditions are central to the success of the method (Pedersen, 2017, Kim et al., 2024).