Robust distribution-free tests for the linear model

Steven MacEachern; Torey Hilbert; Yuan Zhang

REVIEW 2 major objections 6 minor 7 references

Robust distribution-free tests for the linear model

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read RobustPALMRT gives a finite-sample type I error bound for testing a covariate in a linear model, under only exchangeable errors, while allowing robust and quantile regression fits with estimated scale.

desk verdict Robust location extension of PALMRT is solid; the dispersion extension is overclaimed because the theorem's conditions are not verified. read the letter →

arxiv 2501.00583 v1 pith:Q3M5OSND submitted 2024-12-31 stat.ME stat.AP

Torey Hilbert , Steven MacEachern , Yuan Zhang This is my paper · ORCID

classification stat.MEstat.AP MSC 62G1062J0562G35

keywords permutationtestrobustregressionquantilefinite-sampletypeIerrorexchangeabilitypartialcorrelationdispersionLong-COVID

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces RobustPALMRT, a permutation-based test for whether a covariate of interest has any effect on a response after adjusting for control covariates in a linear model. The central theoretical claim is that the test controls the type I error rate in finite samples, with $P(\mathrm{pvalue} \le \alpha) \le 2\alpha$, assuming only that the errors are exchangeable and that the chosen fitting method is shift-invariant and permutation-equivariant. This matters because heavy-tailed, skewed, or outlier-laden noise, common in biological data, inflates false positives for classical tests like the F-test. The paper shows that robust regression with scale estimation fits inside the framework, and that separating model fitting from model evaluation can improve power. It also introduces DispersionPALRMT, a quantile-regression variant that tests for differences in dispersion and finds new immune differences in Long-COVID patients.

What carries the argument

The load-bearing object is the pair-of-permutations comparison array $T(\pi_1,\pi_2;\epsilon) = \omega(M(\epsilon, X_{\pi_2}, [Z_{\pi_1}, Z_{\pi_2}]))$, whose defining symmetry $T(\sigma\circ\pi_1,\sigma\circ\pi_2;\epsilon_\sigma) = T(\pi_1,\pi_2;\epsilon)$ transfers exchangeability of the errors into exchangeability of columns of the $B\times B$ comparison matrix $A_{\pi_1,\pi_2} = I(T(\pi_1,\pi_2;\epsilon)\ge T(\pi_2,\pi_1;\epsilon))$. A weighted-column tournament argument then bounds the probability that the identity column's average is small, giving the $2\alpha$ guarantee. Conditions 1 and 2 on the fitting method $M$ -- shift invariance in the $[Z, Z_\pi]$ directions and joint row-permutation equivariance -- are what let $M(Y, X, [Z, Z_\pi])$ be replaced by $M(\epsilon, X, [Z, Z_\pi])$ under the null, so the proof goes through for robust and quantile regressions rather than only OLS projections.

What would settle it

Simulate the null model $Y = Z\theta + \epsilon$ with exchangeable, symmetric, heavy-tailed errors and an independent case/control label, run DispersionPALRMT at nominal level $\alpha$, and check whether its rejection rate exceeds $2\alpha$; if it does, the grouped residual summaries break the permutation-invariance condition and Theorem 1 does not apply to that procedure.

Watch

Extended reading notes

Core claim

The core discovery is an extension of the permutation-augmented linear regression test (PALMRT) to any fitting method that satisfies two verifiable conditions: adding any shift of the control covariates must not change the output, and permuting the data rows must permute the output in the same way. Under the null model $Y = Z\theta + f(X,\epsilon)$ with exchangeable $\epsilon$, the paper proves that the permutation p-value defined by comparing fits to $(Y, X, [Z, Z_\pi])$ and $(Y, X_\pi, [Z, Z_\pi])$ satisfies $P_{H_0}(\mathrm{pvalue} \le \alpha) \le 2\alpha$. Because robust regression with a Huber loss and MAD scale estimation satisfies the conditions, the result covers robust and quantile regression, including hyper-parameter tuning and scale estimation. The paper further shows empirically that evaluating models with a robust loss improves power under heavy tails and skew even when the model is fit by OLS, and that the quantile-based DispersionPALRMT detects scale differences in the Long-COVID immune data that location-focused tests miss.

Load-bearing premise

The test's guarantee rests on the errors being exchangeable after adjusting for controls, and on the chosen fitting method ignoring shifts in the control covariates and treating all rows symmetrically; the paper verifies these properties for robust regression but flags that the quantile-based dispersion procedure does not clearly satisfy the row-permutation condition.

Editorial extensions

If this is right

Using a level $\alpha/2$ cutoff gives strict finite-sample control at $\alpha$ for any method satisfying the two conditions, even with estimated scale parameters.
Analysts can fit with OLS but evaluate with a robust loss and keep the guarantee while gaining power under heavy-tailed errors; the evaluation choice matters independently of the fit.
DispersionPALRMT provides a finite-sample permutation test for differences in spread between groups, not just location, after adjusting for covariates.
Any future fitting method verified to satisfy shift invariance and permutation equivariance inherits the same type I error bound, so the method is not tied to least squares.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the $2\alpha$ bound is loose in simulations; an adaptive recalibration using the empirical null distribution of p-values, or an exact correction based on the comparison matrix's column sums, could close the gap and restore nominal $\alpha$ without losing the finite-sample guarantee.
Editorial inference: DispersionPALRMT's grouped case/control summaries appear to violate the row-permutation condition, so its guarantee is not covered by Theorem 1; a symmetric dispersion statistic based on all pairwise absolute residual differences might recover the finite-sample control while keeping the power advantage.
Editorial inference: the separation of fit and evaluation suggests a cheap robustness upgrade for existing permutation tests: replace the $L_2$ score with a robust loss, which simulations show can raise power substantially without changing how the model is estimated.
Editorial inference: because the theory only needs exchangeability, extending it to multivariate responses (as the paper notes) and to regularization paths for high-dimensional $X$ is a natural next step; high-dimensional $Z$ remains the known bottleneck.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know about this paper is that the main contribution is solid, but the advertised scope goes beyond what the theory actually covers. The extension of PALMRT to robust regression with scale estimation (the Huber-Huber procedure) is real, useful, and appears to satisfy the conditions of the main theorem. The DispersionPALRMT quantile-regression extension, however, is presented as if it were covered by the same theorem, and it is not. No verification is given that Condition 2 (joint row-permutation invariance) holds for the grouped residual summaries, and on inspection it fails: the grouping is by the same X that gets permuted, so the output changes under a joint row permutation. The finite-sample type I error guarantee therefore does not apply to that procedure.

What is genuinely new: the shift-invariance condition replacing least-squares projection, the proof that Huber regression with MAD scale estimation fits the conditions (Appendix A, Lemma 1), and the finding that separating model fitting from evaluation—specifically using a robust loss in the evaluation step—can substantially improve power. The simulations are thorough, the proof of Theorem 1 is detailed and builds on Guan's framework, and the code is available. The Long-COVID application is a nice demonstration of the location test.

Where I would push back: the abstract says the framework "controls type I error rate for finite-samples" without mentioning the 2α factor. The theorem gives a 2α bound, and the paper itself notes the factor can be necessary in extreme cases. That is a minor overstatement, though the text is honest about it. More seriously, the DispersionPALRMT gap is load-bearing for one of the three advertised contributions. The authors should either prove the conditions for a dispersion-type procedure (perhaps with within-group permutations or a different invariant) or clearly state that the theoretical guarantee does not cover this extension. As it stands, a reader could walk away thinking the type I error of their quantile-regression dispersion test is provably controlled when it is not.

Who is this for: people working on permutation tests, robust regression, and heavy-tailed biological data. The Huber-Huber location test is ready to use; the dispersion test is promising and empirically conservative but needs a theoretical fix or a caveat. I would send this to peer review—it deserves careful referees—but I would expect a major revision before publication.

Referee Report

2 major / 6 minor

Summary. The paper proposes RobustPALMRT, a permutation-based framework for testing whether a covariate X has an effect on a response Y after adjusting for control covariates Z. The main theoretical result, Theorem 1, states that if the model-fitting procedure M satisfies two invariance conditions (shift invariance under adding [Z,Zπ]γ to Y, and joint row-permutation invariance) and the errors are exchangeable, then the resulting p-value satisfies P(pvalue ≤ α) ≤ 2α. The authors verify these conditions for a Huber regression with MAD scale estimation (Appendix A) and use the framework to develop Huber-Huber RobustPALMRT. They also propose DispersionPALRMT, a quantile-regression-based test for differences in dispersion between case and control groups, and apply it to Long-COVID immune profiling data.

Significance. If the central guarantee is correct, Theorem 1 substantially extends the PALMRT framework of Guan (2024) to robust regression with estimated scale, quantile regression, and multiple model fits, while retaining a finite-sample type I error bound under only exchangeability. The separation of model fitting from model evaluation is a useful contribution, and the simulations suggest meaningful power gains under heavy-tailed and skewed errors. The paper provides a full proof of the theorem in Appendix C and a detailed verification for the Huber-Huber procedure, with code and reproducibility information mentioned. However, the advertised scope includes DispersionPALRMT, and the formal verification for that procedure is missing; this is a load-bearing gap that needs to be addressed before the paper can be accepted.

major comments (2)

[Section 4.3 / Algorithm 2] The formal definition of M for DispersionPALRMT does not match the structure required by the framework. The displayed M packs residuals from both fits Y ∼ X+Z+Zπ and Y ∼ Xπ+Z+Zπ into one tuple, whereas the framework comparison (6) evaluates M separately at (Y,X,[Z,Zπ]) and (Y,Xπ,[Z,Zπ]). With the displayed definition, M(Y,Xπ,[Z,Zπ]) would fit Y ∼ Xπ+Z+Zπ and Y ∼ Xππ+Z+Zπ, which is not the intended permuted fit. Algorithm 2 implements the intended comparison with a scalar function of a single fit, but the paper never states M in that form nor verifies Conditions 1 and 2 for it. A proof of shift invariance and joint row-permutation invariance for quantile-regression residual summaries, including a tie-breaking convention for non-unique quantile solutions, is needed before Theorem 1 can be invoked for DispersionPALRMT. Without it, the abstract's finite-sample type I error claim for the dispersion extension is unsupported.
[Section 4.2, non-unique minimizers] The paragraph on non-unique minimizers states that 'standard software packages do this [preserve Condition 1]' without proof. For Theorem 1 to apply to the implemented algorithms, the tie-breaking convention must satisfy both Condition 1 and Condition 2, not merely Condition 1. A convention that selects, for instance, the first optimum in a fixed row order would break joint row-permutation invariance. The authors should specify the exact tie-breaking rule used in Algorithms 1 and 2 (e.g., the R or Python implementation) and verify that it preserves both invariance conditions.

minor comments (6)

[Section 4.3, M definition] The tuple defining M lists 'rqLow,Control' twice and omits 'rqHigh,Control'; it should list both quantiles for both groups, e.g., (r_qLow,Control, r_qHigh,Control, r_qLow,Case, r_qHigh,Case).
[Section 4.3 / Algorithm 2] The notation 1/|Xτ=j| is ambiguous; it should be 1/n_j with n_j = |{i : Xτ_i = j}|, and the set notation should be used consistently.
[Appendix C, Theorem 2] In the statement of Theorem 2, the condition on ̃ω is written as '̃ω(M1, M2) + ̃ω(M2, M2) = 1'; it should be '̃ω(M1, M2) + ̃ω(M2, M1) = 1'.
[Section 2.3] The claim that this is 'the first test for robust regression parameters with scale estimation that is valid in a distribution-free setting without moment assumptions in finite-samples' is strong; the authors should verify this against recent literature (e.g., invariance-based and permutation-based tests) and, if it stands, provide a precise comparison.
[Section 5.2] When reporting the 'highly conservative' type I error of DispersionPALRMT (e.g., 0.004 in the normal-errors setting with n=100), the paper should report Monte Carlo standard errors or confidence intervals so that readers can assess the precision of these empirical rates.
[Section 1, Figure 1] The text 'Figure 1a shows a normal QQ plot ... we see that there is shows strong right-skewness' contains a typo ('there is shows'); it should read 'we see that there is strong right-skewness'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is an honest generalization of external PALMRT results with an independent proof; the DispersionPALRMT gap is a scope/correctness issue, not a circular reduction.

full rationale

The paper's central guarantee, Theorem 1, is not obtained by assuming its own conclusion. The proof reduces to Guan's Theorem 3.1 (an external, published result) and the tournament bound of Barber et al. (2023), and Appendix C supplies a self-contained proof of a more general population version, reproducing the needed combinatorial lemma rather than importing the target result. The two conditions on M are verified for the Huber-Huber procedure in Appendix A: Lemma 1 proves shift invariance and permutation equivariance for Algorithms A1/A2, and the displayed algebra shows these imply Conditions 1 and 2. No fitted parameter is relabeled as a prediction; the 2-alpha bound is not an input to the construction, and no equation is definitionally equivalent to the p-value bound. The only self-citation, Jung, MacEachern and Kim (2021), supports an empirical power observation and is not load-bearing. The quantile-based DispersionPALRMT (Section 4.3, Algorithm 2) is never shown to satisfy Conditions 1 and 2, and Section 8 explicitly concedes that type I error may not be controlled under unmodeled heteroskedasticity; however, this is an unverified applicability claim, not a circular derivation. The central Huber-Huber result is self-contained against external benchmarks, so no significant circularity is present.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central theorem relies on two explicit conditions on the fitting algorithm and two distributional assumptions. The paper verifies these conditions for the Huber regression implementation, but not for the quantile-regression dispersion test, which is the main gap in the ledger.

free parameters (3)

Huber constant delta = 1.345
Standard robust statistics tuning constant in the Huber loss; chosen a priori, not fitted to data. The validity theorem does not depend on its value.
MAD normal-consistency constant = 1.4826
Standard scaling factor in robust regression (Venables & Ripley 2013); fixed constant, not fitted.
Quantile levels for DispersionPALRMT = qLow=0.10, qHigh=0.90
Chosen to define the 80% conditional interquartile range for the dispersion test; arbitrary but fixed, not fitted to achieve a result.

assumptions (5)

domain assumption Assumption 1: ϵ is an exchangeable random vector.
Core distributional assumption for the permutation argument; stated in Section 3.1 and used throughout the proof of Theorem 1.
domain assumption Assumption 2: Y is linear in Z, i.e., Y = Zθ + f(X, ϵ).
The null model restricts the effect of Z to be additive and linear; stated in Section 3.1 and used to eliminate Zθ via shift invariance.
ad hoc to paper Condition 1: M is shift-invariant: M(Y + [Z, Zπ]γ, X, [Z, Zπ]) = M(Y, X, [Z, Zπ]).
Explicitly verifiable condition on the fitting algorithm; verified for the Huber procedure in Lemma 1, not verified for the quantile-regression dispersion procedure.
ad hoc to paper Condition 2: M is invariant to joint row permutations: M(Yσ, Xσ, [Z, Zπ]σ) = M(Y, X, [Z, Zπ]).
Requires the fitting output to be symmetric in cases; holds for sorted residuals, doubtful for grouped summaries in DispersionPALRMT.
standard math Tournament bound (Lemma 2): For a matrix A with Aij + Aji = 1, weighted column sums ≤ α have total weight ≤ 2α.
Combinatorial lemma adapted from Barber et al. (2023); proven in Appendix C.

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Cite this review

Pith. "Pith review of Robust distribution-free tests for the linear model." pith.science (2026). https://pith.science/paper/Q3M5OSND

@misc{pith2026250100583,
  author       = {Pith},
  title        = {Pith review of: Robust distribution-free tests for the linear model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q3M5OSND}},
  note         = {Machine review of arXiv:2501.00583}
}

read the original abstract

Recently, there has been growing concern about heavy-tailed and skewed noise in biological data. We introduce RobustPALMRT, a flexible permutation framework for testing the association of a covariate of interest adjusted for control covariates. RobustPALMRT controls type I error rate for finite-samples, even in the presence of heavy-tailed or skewed noise. The new framework expands the scope of state-of-the-art tests in three directions. First, our method applies to robust and quantile regressions, even with the necessary hyper-parameter tuning. Second, by separating model-fitting and model-evaluation, we discover that performance improves when using a robust loss function in the model-evaluation step, regardless of how the model is fit. Third, we allow fitting multiple models to detect specialized features of interest in a distribution. To demonstrate this, we introduce DispersionPALRMT, which tests for differences in dispersion between treatment and control groups. We establish theoretical guarantees, identify settings where our method has greater power than existing methods, and analyze existing immunological data on Long-COVID patients. Using RobustPALMRT, we unveil novel differences between Long-COVID patients and others even in the presence of highly skewed noise.

Figures

Figures reproduced from arXiv: 2501.00583 by the authors.

**Figure 1.** a shows a normal QQ plot from a fit of model (1), assuming ϵi ’s are i.i.d. normal, for the proportions Y for one type of cell. We see that there is shows strong right-skewness, clearly violating the assumption that the ϵi ’s are normally distributed. In figure 1b we plot the residuals for a fit from model (1) for three different types of cells. We see that the residuals for control patients are mildly skewed, while… view at source ↗

**Figure 2.** Empirical CDF of PALMRT and RobustPALMRT for p-values under H0. If the p-values for a method are calibrated correctly, the simulated curve will lie near or below the dashed line, which is the cumulative distribution function (CDF) of the uniform distribution. Notice that for all nonnormal settings, the F-test spikes far above the uniform CDF for small nominal levels, while all RobustPALMRT methods stay below the uni… view at source ↗

**Figure 3.** Relative power of RobustPALMRT to the the F-test. More specifically, (Power of RobustPALMRT) / (F-test power) vs F-test power. Error bars are provided for the OLS-L2 and Huber-Huber methods, but are very small. The curves in these panels have been “jittered” horizontally to preserve their shapes while enhancing the visualization. The nominal level of the F-test is 0.05. Notice that for all nonnormal settings, the Hu… view at source ↗

Figures from the paper (18 more)

**Figure 4.** Figure 4: Relative power of RobustPALMRT to the the F-test. More specifically, (Power of RobustPALMRT) / (F-test power) vs sample size n, fixing F-test power at 40%. Error bars are provided for the OLS-L2 and Huber-Huber methods, but are very small. The curves in these panels ha…

**Figure 5.** Figure 5: Power of the Breusch-Pagan test and DispersionPALRMT in model (8). The power was computed by 1000 Monte-Carlo replicates at level α = 0.05. Error bars are plotted, but are nearly invisible due to their negligible width. The title of each plot indicates the noise distri…

**Figure 6.** Figure 6: 95% CI’s for the effect of Long-COVID using Huber-Huber (red, left) and OLS-L2 (blue, right). The intervals are scaled by the standard deviation of the cell type proportion (Y (k) ) [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

**Figure 7.** Figure 7: Relative power for various RobustPALRMT regression approaches compared to [PITH_FULL_IMAGE:figures/full_fig_p049_7.png]

**Figure 8.** Figure 8: Relative power for various RobustPALRMT regression approaches compared to [PITH_FULL_IMAGE:figures/full_fig_p050_8.png]

**Figure 9.** Figure 9: Relative power for various RobustPALRMT regression approaches compared to [PITH_FULL_IMAGE:figures/full_fig_p051_9.png]

**Figure 10.** Figure 10: Relative power of RobustPALMRT compared to the F-test vs sample size [PITH_FULL_IMAGE:figures/full_fig_p052_10.png]

**Figure 11.** Figure 11: Relative power of RobustPALMRT compared to the F-test vs sample size [PITH_FULL_IMAGE:figures/full_fig_p053_11.png]

**Figure 12.** Figure 12: Relative power of RobustPALMRT compared to the F-test vs sample size [PITH_FULL_IMAGE:figures/full_fig_p054_12.png]

**Figure 13.** Figure 13: Relative power of RobustPALMRT compared to the F-test vs sample size [PITH_FULL_IMAGE:figures/full_fig_p055_13.png]

**Figure 14.** Figure 14: Relative power of RobustPALMRT compared to the F-test vs sample size [PITH_FULL_IMAGE:figures/full_fig_p056_14.png]

**Figure 15.** Figure 15: Empirical CDF of RobustPALMRT and F-test p-values, with [PITH_FULL_IMAGE:figures/full_fig_p057_15.png]

**Figure 16.** Figure 16: Empirical CDF of RobustPALMRT and F-test p-values, with [PITH_FULL_IMAGE:figures/full_fig_p058_16.png]

**Figure 17.** Figure 17: Empirical CDF of RobustPALMRT and F-test p-values, with [PITH_FULL_IMAGE:figures/full_fig_p059_17.png]

**Figure 18.** Figure 18: Empirical CDF of RobustPALMRT and F-test p-values, with [PITH_FULL_IMAGE:figures/full_fig_p060_18.png]

**Figure 19.** Figure 19: Empirical CDF of RobustPALMRT and F-test p-values, with [PITH_FULL_IMAGE:figures/full_fig_p061_19.png]

**Figure 20.** Figure 20: Empirical CDF of RobustPALMRT and F-test p-values, with [PITH_FULL_IMAGE:figures/full_fig_p062_20.png]

**Figure 21.** Figure 21: Empirical CDF of RobustPALMRT and F-test p-values, with [PITH_FULL_IMAGE:figures/full_fig_p063_21.png]

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Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages

[1]

F., Candès, E

Barber, R. F., Candès, E. J., Ramdas, A. & Tibshirani, R. J. (2021), ‘Predictive inference with the jackknife+’,The Annals of Statistics49(1), 486 –

work page 2021
[5]

• ϵ ∼ N (0, In) + (−1)Bernoulli(0.5) · 104 ·Multinomial 1; 1 n ,

- Log Normal errors. • ϵ ∼ N (0, In) + (−1)Bernoulli(0.5) · 104 ·Multinomial 1; 1 n , . . . ,1 n - Multinomial errors. The Log Normal setting has individual entrieslog(ϵi) ∼ N (0, 1). This is a moderately skewed setting, having skewnessγ1 = 6.18. The Multinomial errors setting represents a Normal errors setting except with one extreme outlier. Lastly, for...

work page 2024
[6]

I X π∈Sn f (π)A(π, τ; ϵ) ≤ α !# = Eϵ

Let Y be generated according to model(3), and letσ, π1, . . . , πB iid ∼ F for any (possibly non- uniform) distribution F over the group of permutationsSn. Then for anyθ ∈ Rp and any exchangeable ϵ, under the null hypothesisH0 we have: PH0 pPop value(σ) ≤ α ≤ 2α, PH0 pMC value(σ) ≤ α ≤ 2α, where the randomness is averaged overϵ, σ, and π1, . . . , πB. If,...

work page 2023
[7]

In order to computeA(π, σ; ϵ) for some σ ̸= Id, we would need to know the true value ofϵ

While it is tempting to try to directly appeal to column exchangeability properties, only the values in the column A(π, Id; ϵ) = A(π; Id; , Y) will be computable from the data. In order to computeA(π, σ; ϵ) for some σ ̸= Id, we would need to know the true value ofϵ. Nonetheless, the Aij + Aji = 1 condition lets us bound how many columns can have small col...

work page 2023
[507]

F., Candes, E

Barber, R. F., Candes, E. J., Ramdas, A. & Tibshirani, R. J. (2023), ‘Conformal prediction beyond exchangeability’,The Annals of Statistics51(2), 816–845. Breusch, T. S. & Pagan, A. R. (1979), ‘A simple test for heteroscedasticity and random coefficient variation’,Econometrica: Journal of the Econometric Societypp. 1287–1294. Cook, R. D. & Weisberg, S. (1...

arXiv 2023
[808]

Wang, L., Peng, B. & Li, R. (2015), ‘A high-dimensional nonparametric multivariate test for mean vector’,Journal of the American Statistical Association110(512), 1658–1669. Wen, K., Wang, T. & Wang, Y. (2022), ‘Residual permutation test for high-dimensional regression coefficient testing’,arXiv preprint arXiv:2211.16182. Young, A. (2024), ‘Asymptotically ...

work page Pith review arXiv 2015
[2013]

hat matrix

Require: Rectangular data set[Y, C]. Y ∈ Rn are responses; C ∈ Rn×r are covariates. R0 ← Residuals from OLS regression ofY on C. while Not converged do sk ← 1.4826 ∗mediani=1,...,n(abs(Rk i )) for i = 1, . . . , ndo wk i ← min 1, 1.345 abs(Rk i )/sk end for Rk+1 ← Residuals from WLS regression ofRk on C with weights wk. Terminate if ||Rk+1−Rk||2 ||Rk||2 <...

work page 2013

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

F., Candès, E

Barber, R. F., Candès, E. J., Ramdas, A. & Tibshirani, R. J. (2021), ‘Predictive inference with the jackknife+’,The Annals of Statistics49(1), 486 –

work page 2021

[2] [5]

• ϵ ∼ N (0, In) + (−1)Bernoulli(0.5) · 104 ·Multinomial 1; 1 n ,

- Log Normal errors. • ϵ ∼ N (0, In) + (−1)Bernoulli(0.5) · 104 ·Multinomial 1; 1 n , . . . ,1 n - Multinomial errors. The Log Normal setting has individual entrieslog(ϵi) ∼ N (0, 1). This is a moderately skewed setting, having skewnessγ1 = 6.18. The Multinomial errors setting represents a Normal errors setting except with one extreme outlier. Lastly, for...

work page 2024

[3] [6]

I X π∈Sn f (π)A(π, τ; ϵ) ≤ α !# = Eϵ

Let Y be generated according to model(3), and letσ, π1, . . . , πB iid ∼ F for any (possibly non- uniform) distribution F over the group of permutationsSn. Then for anyθ ∈ Rp and any exchangeable ϵ, under the null hypothesisH0 we have: PH0 pPop value(σ) ≤ α ≤ 2α, PH0 pMC value(σ) ≤ α ≤ 2α, where the randomness is averaged overϵ, σ, and π1, . . . , πB. If,...

work page 2023

[4] [7]

In order to computeA(π, σ; ϵ) for some σ ̸= Id, we would need to know the true value ofϵ

While it is tempting to try to directly appeal to column exchangeability properties, only the values in the column A(π, Id; ϵ) = A(π; Id; , Y) will be computable from the data. In order to computeA(π, σ; ϵ) for some σ ̸= Id, we would need to know the true value ofϵ. Nonetheless, the Aij + Aji = 1 condition lets us bound how many columns can have small col...

work page 2023

[5] [507]

F., Candes, E

Barber, R. F., Candes, E. J., Ramdas, A. & Tibshirani, R. J. (2023), ‘Conformal prediction beyond exchangeability’,The Annals of Statistics51(2), 816–845. Breusch, T. S. & Pagan, A. R. (1979), ‘A simple test for heteroscedasticity and random coefficient variation’,Econometrica: Journal of the Econometric Societypp. 1287–1294. Cook, R. D. & Weisberg, S. (1...

arXiv 2023

[6] [808]

Wang, L., Peng, B. & Li, R. (2015), ‘A high-dimensional nonparametric multivariate test for mean vector’,Journal of the American Statistical Association110(512), 1658–1669. Wen, K., Wang, T. & Wang, Y. (2022), ‘Residual permutation test for high-dimensional regression coefficient testing’,arXiv preprint arXiv:2211.16182. Young, A. (2024), ‘Asymptotically ...

work page Pith review arXiv 2015

[7] [2013]

hat matrix

Require: Rectangular data set[Y, C]. Y ∈ Rn are responses; C ∈ Rn×r are covariates. R0 ← Residuals from OLS regression ofY on C. while Not converged do sk ← 1.4826 ∗mediani=1,...,n(abs(Rk i )) for i = 1, . . . , ndo wk i ← min 1, 1.345 abs(Rk i )/sk end for Rk+1 ← Residuals from WLS regression ofRk on C with weights wk. Terminate if ||Rk+1−Rk||2 ||Rk||2 <...

work page 2013