Sample-Fitted Stabilized Test Methods

• The study introduces sample-fitted stabilized tests that adjust for sample-induced bias and instability through data-driven transformations and calibrated resampling.
• Methodologies include sample splitting, eigenvalue randomization, and regression-based calibration to maintain Type I error control and enhance power in high-dimensional settings.
• These approaches provide reliable inference across various applications, offering robust diagnostics, optimal calibration, and improved performance in real-world data scenarios.

A sample-fitted stabilized test is a class of methodologies in statistical inference and machine learning that constructs test statistics or predictive estimators using transformations or procedures specifically designed to compensate for instability, bias, and sample size variability inherent in the “fitted” quantities directly computed from observed data. These approaches aim to ensure robust performance, proper error control, or reliable calibration across a range of scenarios such as dependence in test residuals, high dimensionality, non-standard sampling distributions, or model misspecification. The stabilization is sample-adaptive ("fitted")—meaning the transformation, splitting, weighting, or critical value adjustment is informed by properties of the empirical sample or learned estimator rather than relying solely on asymptotics, theory, or precomputed tables.

1. Fundamental Stabilization Principles in Sample-Fitted Testing

Stabilized tests emerge when standard inferential procedures are invalidated by sample-induced effects—such as dependence induced by parameter estimation, high-dimensional noise, or finite-sample distortions of the null distribution. The main principles are:

• Transformation for Null Distribution Invariance: Adjusting a statistic by a data-driven or parameterized transformation so that its (possibly finite-sample) null distribution is stabilized against sample size or estimation-induced artifacts. A quintessential example is the quantile or scaling adjustment to goodness-of-fit test statistics so the critical values become nearly invariant across $n$ (Fernández-de-Marcos et al., 2021).
• Sample Splitting: Partitioning data into estimation and testing subsets to prevent "double-dipping" bias—where model parameters and test statistics both depend on the same data—so the empirical distribution of the stabilized test statistic matches that in an idealized independent-sample world (Davis et al., 11 Mar 2024).
• Statistical Randomization and Double Randomization: Injecting randomization steps to remove or "average out" instability or bias in the null distribution, especially for eigenvalue-based tests where consistent estimation of population quantities is impossible under the null (Barigozzi et al., 2017).
• Model-Adaptive Calibration: Using the empirical sample to compute tailored transformations (e.g., regression-based, isotonic, or monotonic mappings) that “fit” the stabilization map to observed sample size, dimensionality, and estimation error structures (Fernández-de-Marcos et al., 2021, Laan et al., 10 Nov 2024).
• Robustification through Weighting, Subsampling, or Design Adjustment: Modifying the empirical loss or estimator by reweighting samples (to counter collinearity or confounding) or extracting design-balanced subsamples, yielding estimators that resist distributional shift or variance inflation (Shen et al., 2019, Kuang et al., 2020).

These principles enable procedures that maintain desired statistical properties such as valid Type I error, minimax optimal power, or distributional calibration across variable sample configurations and realistic inferential regimes.

2. Methodological Frameworks and Test Construction

Sample-fitted stabilized test construction proceeds through several key frameworks:

• Block Maxima Ratio Tests in Random Fields: Detecting long memory in stable processes using the ratio of maxima over spatially separated blocks, yielding limit distributions insensitive to nuisance parameters and robust to infinite variance (Bhattacharya et al., 2016).
• Eigenvalue-based Monitoring with Double Randomization: Monitoring multivariate time series or high-dimensional factor models for structural breaks by randomizing the statistics computed from sample covariance matrices, producing i.i.d. test statistics with standard null limits even as the sample eigenvalues themselves remain "unstabilized" (Barigozzi et al., 2017).
• Projection and Regression-based Null Calibration: Transforming classical goodness-of-fit statistics (e.g., Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling) using regression-based (“fitted”) parametric functions of $n$ and significance threshold $\alpha$ to deliver correct upper-tail probabilities for arbitrary sample sizes (Fernández-de-Marcos et al., 2021).
• Sample Splitting in Dependent Models: For residual-based goodness-of-fit testing in parametric time series models, divide the data into estimation and evaluation sets; this ensures the empirical autocorrelation and auto-distance correlation statistics converge in distribution to their i.i.d. model-based limits, obviating the need for explicit corrections (Davis et al., 11 Mar 2024).
• Isotonic Calibration of Inverse Probability Weights: In causal inference, replacing raw or inverted propensity scores with data-driven nonincreasing isotonic transformations minimizes instability (especially when propensity scores are near zero), guaranteeing lower calibration error and improved doubly robust estimator performance (Laan et al., 10 Nov 2024).
• Adaptive Combination Tests (Stable Combination Tests): Aggregating $p$-values via functions (e.g., stable law quantile transformations) whose normalization and stability parameters are fitted to the empirical tail or dependence structure, optimizing finite-sample and asymptotic power and error control (Ling et al., 2021).

These frameworks can be algorithmically intricate, involving regression fitting, optimization (e.g., isotonic regression, likelihood maximization with constraints), resampling blocks, functional transformation estimation, and appropriate use of simulations to validate calibration.

3. Theoretical Properties and Statistical Guarantees

Stabilization procedures are supported by rigorous theoretical guarantees tailored to address the empirical instabilities and nonstandard behavior of sample-based procedures. Illustrative results include:

• Null Distribution Stabilization: Under proper transformation, finite-sample or large-sample critical values and $p$-values are proven to have asymptotically correct Type I error, even when the untransformed statistic has a complex or non-pivotal null distribution (Fernández-de-Marcos et al., 2021, Barigozzi et al., 2017).
• Power Consistency: For block maxima and randomization-based change-point tests, the power function converges to one under alternatives as sample size grows, despite the lack of consistent estimators under the null (Bhattacharya et al., 2016, Barigozzi et al., 2017).
• Uniform Validity Across Sample Sizes: Regression-based stabilization maps ensure that critical values and $p$-values are properly calibrated for a continuum of $(n, \alpha)$, enabling valid inference in sequential and online settings (Fernández-de-Marcos et al., 2021).
• Robustness to Estimation-induced Dependence: Sample splitting in parametric residual analysis ensures that, with appropriate overlapping fraction, the limit variance of residual autocorrelation statistics matches the theoretical i.i.d. model, restoring classical null distributions without further adjustment (Davis et al., 11 Mar 2024).
• Calibration Error and Efficiency Bounds: For isotonic-calibrated inverse probability weights, the calibration error converges to zero at a rate $O_p(n^{-2/3})$ and the mean squared error is nearly globally optimal over the class of monotone transformations, resulting in doubly robust estimators with the standard efficiency properties (Laan et al., 10 Nov 2024).
• Non-distinguishability in Fitted Models: When fitting a flexible model (e.g., negative binomial to Poisson data), as the sample size increases the fitted parameters can converge to boundary values corresponding to the null, and goodness-of-fit statistics (e.g., Kolmogorov–Smirnov) become unable to distinguish over-parametrization, highlighting both stabilization and its diagnostic limitations (Yang et al., 11 Apr 2024).

These results underpin the rationale and practical significance of stabilization: without adjustment, empirical tests can be either too conservative, anti-conservative, or uncalibrated when standard inferential theory is naively applied.

4. Practical Applications and Empirical Performance

Stabilized tests find broad application across diverse statistical and machine learning domains:

Problem Area	Stabilization Principle	Key Benefits
Goodness-of-fit for distributions	Regression-based critical value adjustment (Fernández-de-Marcos et al., 2021)	Rapid, accurate $p$-values for small $n$; sequential testing
High-dimensional factor model monitoring	Double randomization of eigenvalue statistics (Barigozzi et al., 2017)	Controlled Type I error, robust sequential change detection
Time series residual analysis	Sample splitting (Davis et al., 11 Mar 2024)	IID-like limit for ACF/ADCF; simple, interpretable diagnostics
Hypothesis testing combination	Stable law–based $p$-value aggregation (Ling et al., 2021)	Flexible error/power trade-off under dependence
Causal inference weighting	Isotonic calibration of IPW (Laan et al., 10 Nov 2024)	Well-calibrated, stable weights; improved ATE estimator coverage
Model selection for count data	Constrained MLE in NB–Poisson setting (Yang et al., 11 Apr 2024)	Prevents degenerate fits, clarifies identifiability limits

Empirical studies corroborate these advantages: for instance, regression-based stabilized goodness-of-fit procedures achieve accurate calibration for $n$ as small as five and outperform traditional Monte Carlo in speed and reliability (Fernández-de-Marcos et al., 2021); isotonic-calibrated inverse probability weights yield marked improvements in RMSE and bias, especially in limited overlap settings (Laan et al., 10 Nov 2024).

5. Limitations, Diagnostics, and Cautions

While sample-fitted stabilization enables robust inference in challenging situations, several methodological caveats and diagnostic themes emerge:

• Potential Overfitting to Sample: In small samples or particularly ill-conditioned regimes, excessively adaptive calibration may compromise inference by fitting noise rather than signal, especially if the stabilization function is not sufficiently regularized.
• Detection Limits under Overparameterization: When models are so flexible that fitted statistics under the null (e.g., NB vs Poisson) become arbitrarily close, goodness-of-fit tests may lose utility, and alternative identification or model complexity penalties may be needed (Yang et al., 11 Apr 2024).
• Boundary Issues in Regression-based Transforms: At the edge of the predictor space, regression- or isotonic-fitted stabilization functions may behave pathologically; adaptive truncation or tailored loss functions are required to control extremal weights or transformed statistics (Laan et al., 10 Nov 2024).
• Assumption Sensitivity: Some stabilization methods (e.g., in time series) require structural assumptions (e.g., stationarity, invertibility, or specific parameter splitting fractions) to achieve the desired asymptotic behaviors (Davis et al., 11 Mar 2024).
• Computational Cost: While regression-based or resampling-based stabilization is often computationally efficient, permutation-based or highly adaptive sample-fitted approaches can be costly in high dimension or large $n$ and may require further methodological innovations for scalability.

A plausible implication is that with increasing data complexity and model adaptivity, routine use of sample-fitted stabilization (with built-in diagnostics for overfit or misspecification) will become necessary in both inferential and machine learning practice.

6. Future Directions and Theoretical Developments

Emerging research avenues in sample-fitted stabilization include:

• Extending Regression-based Stabilization to Multisample and Complex Data Structures: Adapting flexible regression frameworks to cover multivariate, structured, or irregularly sampled data is an open problem with high impact (Fernández-de-Marcos et al., 2021).
• Unifying Finite-Sample and Asymptotic Calibration: Generalizing stabilization beyond upper-tail critical values to entire null distributions, yielding fully uniform $p$-values under the null across sample sizes and parameter regimes (Fernández-de-Marcos et al., 2021).
• Sample-Adaptive Combination Tests: Systematically learning the parameters of the stabilizing transformation (e.g., stable law exponent, weight vector) to optimize test power under empirical dependence, potentially via cross-validation or bootstrapping (Ling et al., 2021).
• Diagnostics for Stabilization Validity: Integrating diagnostic checks for the appropriateness of the fitted stabilization, e.g., via bootstrap, sample splitting, or high-dimensional noise detection.
• Applications in online, sequential, and adaptive learning: Expanding stabilized test frameworks to online change detection, streaming data, and adaptive model selection with automatic, sample-fitted calibration.

The continued theoretical and computational development of sample-fitted stabilized tests promises to supply the statistical community with tools for principled, reliable inference in scenarios where classical approaches fail or require ad hoc adjustments.