Better Statistical Tests Minimise False Error Rates

Researchers are increasingly focused on refining goodness-of-fit (GOF) testing procedures, and Nicholas G. Polson from Booth School of Business, University of Chicago, Vadim Sokolov from Volgenau School of Engineering, George Mason University, and Daniel Zantedeschi from School of Information Systems, Muma College of Business, University of South Florida, present a novel framework utilising Bayes risk. Their collaborative work establishes that optimal calibration for these tests operates on a moderate-deviation scale, offering a new perspective on Type I error control and canonical inflation of rejection. This research formalises the Rubin, Sethuraman program for Kolmogorov, Smirnov-type statistics as a risk-calibration process, connecting Bayes-risk expansions with Sanov information asymptotics and providing applications to location and shape testing, ultimately unifying existing perspectives under a single, coherent risk criterion.

Scientists have long sought better ways to assess how well statistical models reflect real-world data. Current methods often struggle with the balance between flagging false positives and missing genuine patterns. A Bayesian approach to goodness-of-fit testing offers a more principled way to make these decisions, improving the reliability of statistical inference.

This work redefines the calibration of goodness-of-fit tests, moving beyond traditional methods reliant on fixed significance levels or large-deviation approximations. It establishes that optimal calibration, when viewed through the lens of Bayes risk, operates on a moderate-deviation scale, fundamentally altering how statistical tests balance false positives and missed detections.

Rather than controlling the Type I error rate at a constant level, this new framework accounts for the increasing number of distinguishable alternatives as sample size grows, leading to a more nuanced and effective approach to model validation. The research details how sample size influences optimal cutoffs for these tests, revealing a unifying principle applicable across various statistical procedures.

Classical goodness-of-fit procedures, including the Kolmogorov-Smirnov test, typically assess how well observed data align with a proposed distribution using either fixed significance levels or exponential error exponents. However, these approaches fall short when the goal is to minimise overall risk, the combined cost of incorrect decisions. Fixed-α calibration disregards the increasing ability to differentiate between distributions as data accumulate, while large-deviation calibration can be overly strict, sacrificing the power to detect realistic alternatives.

This study demonstrates that a Bayes-risk approach naturally aligns with a moderate-deviation scale, a regime between the extremes of fixed-α and large-deviation methods. At the heart of this work lies a formalisation of the Rubin-Sethuraman program for KS-type statistics, establishing explicit conditions for prior distributions and empirical-process functionals.

Researchers connected Bayes-risk expansions with Sanov information asymptotics, revealing how truncations arise when risk, rather than just exponents, is the primary evaluation criterion. Detailed applications to location testing with Laplace families, shape testing using Bayes factors, and connections to Fisher information geometry further illustrate the versatility of this new framework.

The organising principle throughout is that sample size impacts Bayes-optimal cutoffs through the moderate-deviation scale, harmonising both KS-based and Sanov-based perspectives under a unified risk criterion. Specifically, the study shows that the optimal rejection threshold for goodness-of-fit tests inflates proportionally to the square root of the logarithm of the sample size, √ log n, and that Type I error decays polynomially.

This scaling arises because the number of distinguishable alternatives grows polynomically with sample size, demanding a risk criterion that accounts for all of them. This perspective builds upon extensive information-theoretic foundations, including Sanov’s work on large deviations and Bahadur’s contributions to moderate-deviation theory. The research provides a reusable calibration principle applicable to a broad range of statistics, including empirical processes and divergence-based procedures, offering a structural consequence of risk balancing rather than a property specific to Bayesian hypothesis testing.

Moderate deviation scaling unifies Bayes-risk optimal goodness-of-fit testing

Initial analyses reveal a clear pattern of moderate deviation scaling in Bayes-risk optimal goodness-of-fit testing. Calibrations operate on a moderate deviation scale, leading to polynomially decaying Type I error, a departure from traditional fixed significance level or large deviation approaches. The research does not rely on Bayes factors or likelihood-ratio structure; instead, Theorem 2.6 and Lemma 2.8 provide a reusable calibration principle applicable to a broad class of statistics, including goodness-of-fit tests and empirical processes.

This principle emerges as a structural consequence of risk balancing, rather than a property specific to Bayesian hypothesis testing. Detailed applications to location testing under Laplace families and shape testing via Bayes factors demonstrate the versatility of this approach. At the heart of the work lies a generic risk decomposition template based solely on Gaussian-type null tails and local prior mass exponents.

This decomposition allows for a precise connection between Bayes-risk expansions and Sanov information asymptotics, showing how order truncations arise naturally when risk, rather than pure exponents, is the evaluation criterion. Once established, this connection clarifies the geometric foundation of the moderate-deviation boundary and connects to Fisher information geometry, providing a deeper understanding of the underlying principles.

Numerical verification of the calibration template confirms the theoretical findings. These results do not depend on specific distributional assumptions, holding for binomial and parametric geometric instances. By isolating this distinct mechanism, the research offers a new perspective on Bayesian calibration and its relationship to information theory.

Moderate-deviation calibration via Bayes risk and Sanov asymptotics

A Bayes risk framework underpins the development of a new approach to goodness-of-fit testing. Rather than relying on traditional calibration methods fixed at a significance level or based on exponential error exponents, this work establishes that optimal calibration occurs on a moderate-deviation scale. The methodology extends beyond adapting existing techniques; it reveals a precise connection between Bayes-risk expansions and Sanov information asymptotics.

Sanov’s theorem describes the rate at which the probability of observing an empirical measure far from the true distribution decays. This connection demonstrates how truncations of order, arising from the risk evaluation criterion, emerge naturally when prioritising risk minimisation. Detailed applications explore location testing with Laplace distributions, shape testing using Bayes factors, and connections to Fisher information geometry.

The experimental setup focuses on understanding how sample size influences Bayes-optimal goodness-of-fit cutoffs. The research demonstrates that sample size enters these cutoffs through the moderate-deviation scale, effectively unifying both Kolmogorov-Smirnov-based and Sanov-based perspectives under a single risk criterion. By servicing all distinguishable alternatives, the methodology avoids overly stringent rejection thresholds that can limit power against realistic, moderate deviations from the null.

Inside this framework, the study leverages the properties of the Fisher distance, quantifying the dissimilarity between probability distributions, to define a boundary shell around the null hypothesis. Distributions within this shell are considered statistically indistinguishable given the sample size, and the prior distribution is constrained to place polynomial mass within it.

This constraint, combined with a careful balancing of false rejections and missed detections, leads to the derived moderate-deviation scaling and polynomial decay of Type I error. The work’s strength lies in its ability to provide a reusable calibration principle applicable to a broad class of statistical tests, including goodness-of-fit tests, empirical processes, and divergence-based procedures.

Balancing statistical risk through moderate deviation testing

Scientists have long sought ways to reliably distinguish between random chance and genuine patterns in data, a problem at the heart of statistical testing. Approaches to ‘goodness-of-fit’, determining how well data align with a proposed model, have relied on either fixed thresholds or extremely conservative error bounds. A new framework proposes a middle ground, calibrating tests to a ‘moderate-deviation’ scale that offers a more balanced approach to risk.

This isn’t merely a technical refinement; it represents a shift in how we conceptualise statistical certainty. The difficulty lies in the inherent tension between avoiding false positives and detecting true effects. Traditional methods often err on one side or the other, either missing real signals or raising alarms over noise. By focusing on Bayes risk, the expected loss from making an incorrect decision, researchers have developed a system where the sample size dictates the precision of the test, unifying previously disparate approaches.

Perspectives based on the Kolmogorov-Smirnov test and Sanov’s theorem now converge under this single risk criterion, offering a more coherent picture. Applying this framework isn’t without its challenges. The calculations involved require careful consideration of prior probabilities and the underlying structure of the data. Determining the appropriate ‘prior’, representing pre-existing knowledge, remains a subjective exercise.

Numbers presented reveal a clear divergence from conventional thresholds, particularly with larger datasets and more complex models, suggesting a need to reassess established practices. However, the implications extend beyond purely academic concerns. Accurate goodness-of-fit testing is vital in fields ranging from medical diagnostics to financial modelling, where misinterpreting data can have serious consequences.

By providing a more nuanced understanding of statistical risk, this work could lead to more reliable and informative analyses. Future research will likely focus on developing practical tools for implementing this framework, exploring its performance in real-world scenarios, and extending it to even more complex statistical problems, potentially reshaping how we validate models across diverse scientific disciplines.