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Inference for quantile-parametrized families via CDF confidence bands

This paper proposes a novel, assumption-lean inference framework that constructs confidence sets for quantile-parametrized families by inverting distribution-free empirical CDF bands, thereby overcoming the computational and theoretical limitations of likelihood-based methods for models lacking closed-form density expressions.

Original authors: Srijan Chattopadhyay, Siddhaarth Sarkar, Arun Kumar Kuchibhotla

Published 2026-08-25

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CC BY 4.0

Original authors: Srijan Chattopadhyay, Siddhaarth Sarkar, Arun Kumar Kuchibhotla

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of statistics, scientists often try to describe the shape of real-world data using mathematical models. Imagine a mapmaker trying to draw a coastline; they need a set of rules to describe how the land meets the sea. In statistics, these rules are called parametric families. For a long time, the most common way to define these shapes was by looking at the density of the data, which is like measuring how crowded the points are at every location. However, some of the most flexible and useful maps cannot be drawn this way because the math for their density is too messy to write down in a simple formula. Instead, statisticians use a different tool: the quantile function. Think of this as a way to describe the data by asking, "Where does the 10th percent of the data end?" or "Where is the middle?" This approach allows for incredibly complex shapes that can mimic everything from smooth hills to jagged spikes, but it creates a new problem. Because the standard density formulas are missing, the usual tools for checking if these models are correct often break down. They can give answers that are wrong, or they can be so unstable that a tiny change in the data leads to a completely different conclusion.

Researchers Srijan Chattopadhyay, Siddhaarth Sarkar, and Arun Kumar Kuchibhotla have developed a new way to solve this problem. They created a method to build reliable confidence sets, which are like safety nets that tell us how sure we can be about the parameters defining our data model. Instead of trying to force the messy density formulas to work, their approach flips the problem around. They start with a known, robust way to create a "confidence band" around the data's cumulative distribution function. This band is a range that is guaranteed to contain the true underlying curve of the data, no matter what that curve looks like. The researchers then take this band and push it through the known quantile function of their model. By doing this, they can see which specific parameter values are consistent with the safe range of the data. It is a direct inversion: if a parameter value would produce a curve that falls outside the safe band, that value is rejected. If it stays inside, it is kept as a possible answer. This method requires no complex assumptions about how the data behaves in the long run and avoids the computational headaches that have plagued previous attempts to analyze these specific types of distributions.

The team tested this framework on two important families of distributions: the Tukey Lambda distribution and the Generalized Lambda distribution. These are famous in the field for their ability to model a vast array of shapes, from symmetric bell curves to highly skewed data with heavy tails. The Tukey Lambda is particularly tricky because its behavior changes drastically depending on its parameter; in some cases, it looks like a normal curve, while in others, it has tails so heavy that standard statistical rules fail. The researchers found that their new method worked beautifully across all these different scenarios. In computer simulations, they showed that their confidence intervals maintained the correct coverage, meaning that if they claimed to be 95% sure, the true value was actually inside the range 95% of the time. This held true even for small sample sizes where other methods, like those relying on bootstrapping or matching quantiles, often failed or produced intervals that were too wide to be useful. The new method consistently produced narrower, more precise intervals while still guaranteeing that the true answer was captured.

To prove the method works in the real world, the authors applied it to two very different datasets. The first was a small sample of birth weights from a study of twins, containing just 123 observations. In this setting, where data is scarce and uncertainty is high, the researchers' method produced a confidence region for the shape of the distribution that was distinct from what standard bootstrap methods suggested. The bootstrap method, which relies on resampling the data over and over, produced a symmetric shape that the new method did not. This suggests the new approach is capturing subtle details about the distribution's shape that the older method missed, providing a more accurate picture of the data's true nature. The second test involved a massive dataset of over 23,000 Spanish household incomes from the 1980s. Here, the data was right-skewed and peaked, a classic challenge for statistical modeling. The new method successfully generated confidence intervals for the location and scale of the income distribution, as well as a joint region for the shape parameters. In this large-sample case, the results aligned with the simulation findings, showing that the method scales up effectively and produces tight, reliable bounds.

The core achievement of this work is that it provides a principled, assumption-lean way to do inference for models that were previously too difficult to handle with standard tools. By relying on the duality between the confidence bands of the data and the quantile function of the model, the researchers bypassed the need for closed-form density expressions and the unstable asymptotic behavior that often plagues these families. They demonstrated that it is possible to get valid, finite-sample guarantees without needing to know exactly where the true parameter lies within the space of possibilities. While the authors note that there is still work to be done, such as refining how to select the specific points used to define the shape parameters in the Generalized Lambda distribution, their framework offers a solid foundation. It turns a class of distributions that were often treated as statistical black boxes into ones that can be analyzed with clarity and precision, ensuring that when scientists map these complex shapes, they know exactly how reliable their map is.

Technical Summary: Inference for Quantile-Parametrized Families via CDF Confidence Bands

Problem Statement
Quantile-based parametric families (e.g., Tukey Lambda, Generalized Lambda Distribution) are powerful tools for modeling data with diverse shapes and tail behaviors using few parameters. However, these models present significant challenges for classical statistical inference. Unlike standard parametric families, their cumulative distribution functions (CDFs) and probability density functions (PDFs) often lack closed-form expressions. This absence prevents the direct application of standard maximum likelihood estimation (MLE) and renders the resulting estimators' asymptotic behavior irregular. Specifically, convergence rates may deviate from the standard $\sqrt{n}$ rate, and limiting distributions can be non-normal or depend heavily on specific parameter values (e.g., near $\lambda=0$ in Tukey Lambda). Consequently, traditional methods like the method of moments, likelihood-based approaches, and even bootstrap/resampling techniques often fail to provide reliable finite-sample coverage guarantees or yield non-standard asymptotics.

Methodology
The authors propose a novel inference framework that bypasses the need for density functions or likelihood ratios by leveraging the duality between confidence bands for the CDF and confidence sets for the parameters of quantile-based families.

Core Framework:
- Assume i.i.d. data $X_1, \dots, X_n$ from a distribution $F_{\Lambda_0}$ with a known quantile function $Q_{\Lambda_0}(p)$ but unknown CDF $F_{\Lambda_0}$ .
- Construct a distribution-free confidence band $[\ell_{n,\alpha,i}, u_{n,\alpha,i}]$ for the CDF such that $\ell_{n,\alpha,i} \le F_{\Lambda_0}(X_{(i)}) \le u_{n,\alpha,i}$ holds with probability $1-\alpha$ for all order statistics $X_{(i)}$ .
- Invert these inequalities through the quantile function to define the confidence set for the parameter $\Lambda$ :
  $\widehat{CI}_{n,\alpha} := \left\{ \Lambda : Q_{\Lambda}(\ell_{n,\alpha,i}) \le X_{(i)} \le Q_{\Lambda}(u_{n,\alpha,i}), \forall i \in [n] \right\}$
- This construction guarantees finite-sample coverage $P_{\Lambda_0}(\Lambda_0 \in \widehat{CI}_{n,\alpha}) \ge 1-\alpha$ without requiring knowledge of the sampling distribution of the estimator.
Confidence Band Selection:
The framework is agnostic to the specific band used but evaluates two primary candidates:
- DKW (Dvoretzky-Kiefer-Wolfowitz): A fixed-width band based on the classic inequality.
- DW (Dümbgen-Wellner): A variable-width band based on a modified Berk-Jones statistic using adjusted KL divergence. The authors demonstrate that the DW band achieves rate-optimal performance, particularly offering tighter bounds at the distribution tails compared to DKW.
Application to Specific Families:
- Tukey Lambda Distribution: The authors analyze the monotonicity of the quantile function with respect to the shape parameter $\lambda$ . They propose a data transformation ( $x \mapsto |x|$ ) to exploit the distribution's symmetry, which yields a sufficient statistic and results in bounded, narrower confidence intervals compared to using raw data.
- Generalized Lambda Distribution (GLD): To handle the four-parameter GLD (location, scale, and two shape parameters), the authors utilize the CSW reparameterization (Chalabi et al., 2012). This separates parameters into:
  - Location ( $\tilde{\mu}$ ) and Scale ( $\tilde{\sigma}$ ): Estimated directly by inverting the CDF band at specific quantiles (median and interquartile range).
  - Shape ( $\chi, \xi$ ): Estimated via normalized shape statistics (ratios of quantile differences). The authors investigate the selection of quantile pairs $(u_1, u_2)$ used to constrain the shape parameters, finding that extreme order statistics (tail behavior) provide the most informative constraints.

Key Results

Simulation Studies (Tukey Lambda): The proposed method maintains nominal coverage (95%) across a wide range of parameter values ( $\lambda \in [-2, 2]$ ) and sample sizes. In contrast, competing methods (L-moments, quantile matching with parametric/non-parametric bootstrap) exhibit significant under-coverage, particularly for $\lambda \le -1$ or near $\lambda = 0$ , where asymptotic irregularities are most pronounced. The DW band consistently produces narrower confidence intervals than the DKW band. In finite samples, the proposed method yields narrower confidence intervals for all $\lambda_0 \ge 1$ .
Simulation Studies (GLD): The method produces valid confidence intervals for location and scale parameters and joint confidence regions for shape parameters. The volume of the confidence regions decreases as sample size increases, consistent with theoretical expectations.
Real Data Applications:
- Twin Study (Small Sample, $n=123$ ): The method produced confidence regions for GLD parameters that differed in shape from bootstrap regions, capturing more distributional information. While the regions were slightly larger than bootstrap regions, this is attributed to the method's strict finite-sample coverage guarantees, whereas bootstrap methods often fail to achieve target coverage in small samples.
- Spanish Household Incomes (Large Sample, $n=23,972$ ): The method successfully modeled the right-skewed, leptokurtic income data. In this large-sample setting, the proposed CDF-based confidence regions converged to be similar in size and shape to bootstrap regions, aligning with simulation trends.

Significance and Claims
The paper claims to provide a "principled and assumption-lean alternative" for inference in complex parametric families where likelihood-based methods are computationally intractable or theoretically unreliable. By relying solely on the known quantile function and distribution-free CDF bands, the framework avoids the computational difficulties of MLE and the theoretical pitfalls of irregular asymptotics. The authors emphasize that their approach offers finite-sample coverage guarantees, a feature often lacking in bootstrap and resampling techniques for these specific distributions. The study demonstrates that this inversion method is robust across varying tail behaviors and sample sizes, offering a practical solution for quality control, finance, and bioassay modeling where quantile-based families are prevalent.

Technical Summary: Inference for Quantile-Parametrized Families via CDF Confidence Bands

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