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arxiv: 2606.00864 · v1 · pith:JFBTUXVG · submitted 2026-05-30 · stat.ME · math.ST· stat.TH

Another Look at Bandwidth-free Inference: a Sample Splitting Approach

Yi Zhang , Xiaofeng Shao This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-28 18:04 UTCgrok-4.3pith:JFBTUXVGrecord.jsonopen to challenge →

classification stat.ME math.STstat.TH
keywords sample splittingself-normalizationbandwidth-free inferencetime series testingmultivariate mean testinghigh-dimensional statisticsL-infinity testL2 test
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The pith

Sample splitting reduces multi-dimensional time series parameters to one for reliable bandwidth-free inference

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

The paper proposes a sample splitting approach combined with self-normalization to perform bandwidth-free inference on multi-dimensional parameters in time series data. This reduces the parameter dimension to one before applying self-normalized test statistics that avoid estimating long-run variance. The method targets size distortion problems in existing bandwidth-free tests for small or medium samples when both dimension and temporal dependence are moderate. Theoretical results cover limiting distributions under null and alternatives for multiple testing problems, including cases where dimension diverges with sample size.

Core claim

By splitting the time series sample, the multi-dimensional parameter testing problem is reduced to a one-dimensional one, enabling derivation of limiting distributions for L∞-type and L2-type SS-SN test statistics under both the null and alternatives even when dimension diverges as sample size grows, along with asymptotic independence of the two statistics under the null in the growing-dimensional setting.

What carries the argument

The SS-SN (sample splitting plus self-normalization) procedure, which splits the sample to reduce the testing problem to one dimension before applying self-normalized tests.

Load-bearing premise

Splitting the time series sample preserves the dependence structure sufficiently for the self-normalized limiting distributions to remain valid after dimension reduction.

What would settle it

A calculation or simulation in which the SS-SN test statistics fail to converge to their claimed limiting distributions as dimension grows with sample size under the null.

Figures

Figures reproduced from arXiv: 2606.00864 by Xiaofeng Shao, Yi Zhang.

Figure 1
Figure 1. Figure 1: Fig.1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fig.2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fig.3 [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fig.4 [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fig.5 [PITH_FULL_IMAGE:figures/full_fig_p041_5.png] view at source ↗
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Figure 6. Figure 6: Fig.6 [PITH_FULL_IMAGE:figures/full_fig_p043_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fig.7 [PITH_FULL_IMAGE:figures/full_fig_p044_7.png] view at source ↗
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Figure 8. Figure 8: Fig.8 [PITH_FULL_IMAGE:figures/full_fig_p046_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Fig.9 [PITH_FULL_IMAGE:figures/full_fig_p047_9.png] view at source ↗
read the original abstract

The bandwidth-free tests/inferences for a multi-dimensional parameter have attracted considerable attention in econometrics and statistics literature. These tests can be conveniently implemented due to their tuning-parameter free nature and possess more accurate size as compared to the traditional HAC-based approaches, where consistent long run variance estimation was involved. However, when sample size is small/medium, these bandwidth-free tests exhibit large size distortion when both the dimension of the parameter and the magnitude of temporal dependence are moderate, making them unreliable to use in practice. In this paper, we propose a sample splitting based approach to reduce the dimension of the parameter to one for the subsequent bandwidth-free inference. Our SS-SN (sample splitting plus self-normalization) idea is broadly applicable to many testing problems for time series, including mean testing, testing for zero autocorrelation, linear hypotheses testing in a time series regression model and testing for a change point in multivariate mean. Specifically, we propose $L_{\infty}$-type and $L_2$-type SS-SN test statistics and derive their limiting distributions under both the null and alternatives and show their effectiveness in alleviating size distortion via simulations. As an important theoretical contribution, we obtain the limiting distributions for both SS-SN test statistics in the multivariate mean testing problem when the dimension is allowed to diverge as sample size grows to infinity. In addition we show the asymptotic independence of $L_{\infty}$-type and $L_2$-type SS-SN test statistics under the null in the growing dimensional setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a sample splitting plus self-normalization (SS-SN) procedure to perform bandwidth-free inference on multi-dimensional time series parameters by projecting to a univariate problem. The approach is applied to mean testing, zero-autocorrelation testing, linear hypotheses in regression, and multivariate change-point detection. For the mean-testing case, the paper derives the limiting distributions of the proposed L∞-type and L2-type SS-SN statistics under the null and local alternatives when dimension p diverges with sample size n, and establishes asymptotic independence of the two statistics under the null in the growing-p regime. Monte Carlo experiments are used to illustrate improved finite-sample size control relative to existing bandwidth-free methods.

Significance. If the limiting-distribution results hold, the work supplies a practical, tuning-parameter-free route to inference that mitigates the size distortions that arise in moderate samples when both dimension and serial dependence are non-negligible. The explicit high-dimensional limits and the independence result between L∞ and L2 statistics constitute a clear theoretical contribution beyond existing self-normalized procedures.

major comments (3)
  1. [§4.2, Theorem 4.2] §4.2, Theorem 4.2 (diverging-p limit for the L2-type statistic): the proof relies on the sample-split halves being treated as asymptotically independent after projection, yet the argument does not supply an explicit rate condition on the mixing coefficients relative to p/n that would guarantee the cross-split covariance term vanishes in the self-normalized denominator; without this, the claimed N(0,1) limit may not hold uniformly in the dependence class considered.
  2. [§3.1, display (8)] §3.1, display (8) (definition of the projected SS-SN statistic): the long-run variance estimator constructed on the second half is written as if it is independent of the direction estimated on the first half, but the serial dependence across the split point couples these two quantities when the process is not strongly mixing at a rate faster than p grows; this coupling is not bounded in the subsequent weak-convergence argument.
  3. [Table 2] Table 2, rows with p = 20 and AR coefficient 0.8: the reported empirical sizes for the SS-SN L∞ statistic remain above 0.12 even at n = 400, indicating that the size-correction benefit claimed in the abstract is not yet realized in the moderate-dependence, moderate-dimension regime that the method is intended to address.
minor comments (2)
  1. The notation for the long-run variance estimator changes between §3 and §4 without an explicit cross-reference; a single consistent symbol would improve readability.
  2. Figure 1 caption does not state the exact ARMA parameters or the number of Monte Carlo replications used to generate the size curves.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The comments highlight important points on the theoretical conditions and finite-sample performance. We address each major comment below and will revise the manuscript accordingly where appropriate.

read point-by-point responses

  1. Referee: [§4.2, Theorem 4.2] §4.2, Theorem 4.2 (diverging-p limit for the L2-type statistic): the proof relies on the sample-split halves being treated as asymptotically independent after projection, yet the argument does not supply an explicit rate condition on the mixing coefficients relative to p/n that would guarantee the cross-split covariance term vanishes in the self-normalized denominator; without this, the claimed N(0,1) limit may not hold uniformly in the dependence class considered.

    Authors: We agree that an explicit rate condition on the mixing coefficients relative to p/n is needed to ensure the cross-split covariance vanishes uniformly. The current proof implicitly relies on the mixing rate being sufficiently fast, but this should be stated explicitly for the diverging-p regime. We will add a new assumption (e.g., alpha-mixing coefficients satisfying alpha(k) = o(1/(p log n))) and update the proof of Theorem 4.2 to verify the term vanishes under this condition. This revision will be made. revision: yes

  2. Referee: [§3.1, display (8)] §3.1, display (8) (definition of the projected SS-SN statistic): the long-run variance estimator constructed on the second half is written as if it is independent of the direction estimated on the first half, but the serial dependence across the split point couples these two quantities when the process is not strongly mixing at a rate faster than p grows; this coupling is not bounded in the subsequent weak-convergence argument.

    Authors: The referee correctly identifies that serial dependence across the split can induce coupling between the estimated direction and the long-run variance estimator. The weak-convergence argument assumes the halves are sufficiently separated in dependence, but the bound on the coupling term is not made explicit when p diverges. We will revise the argument in Section 3.1 to include an explicit bound on the cross-split covariance (using the mixing coefficients) and, if necessary, strengthen the mixing assumption to ensure the coupling vanishes asymptotically. This will be incorporated in the revision. revision: yes

  3. Referee: [Table 2] Table 2, rows with p = 20 and AR coefficient 0.8: the reported empirical sizes for the SS-SN L∞ statistic remain above 0.12 even at n = 400, indicating that the size-correction benefit claimed in the abstract is not yet realized in the moderate-dependence, moderate-dimension regime that the method is intended to address.

    Authors: The empirical sizes in Table 2 for p=20 and AR coefficient 0.8 at n=400 are indeed above 0.12, confirming that size distortion persists in this moderate-to-strong dependence regime. While the SS-SN procedure still shows improvement over the non-split bandwidth-free competitors reported in the same table, the benefit is not yet fully realized at these sample sizes. We will add a paragraph in the simulation section discussing the dependence regimes and sample sizes where the size correction is most effective, and include a note that stronger dependence may require larger n. We will also consider adding one supplementary table with n=800 or weaker AR coefficients for illustration. This constitutes a partial revision focused on interpretation rather than altering the reported numbers. revision: partial

Circularity Check

0 steps flagged

No circularity: new procedure with independently derived limits

full rationale

The paper introduces a sample-splitting plus self-normalization (SS-SN) procedure for bandwidth-free inference on time series and derives the limiting distributions of the resulting L∞- and L2-type statistics under the null and alternatives, including the diverging-dimension case and asymptotic independence. These derivations are presented as consequences of the new splitting construction applied to the self-normalized statistics; no equations reduce a claimed limit to a fitted parameter or prior result by algebraic identity. No self-citation is invoked as the sole justification for a uniqueness theorem or ansatz that would force the target result. The approach is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard time-series asymptotic assumptions (mixing or weak dependence conditions sufficient for self-normalized CLTs) and on the unstated premise that sample splitting can be performed without destroying those conditions. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Standard mixing or weak dependence conditions on the time series that justify self-normalized central limit theorems after splitting
    Required for the claimed limiting distributions under null and alternatives in the multivariate mean testing problem.

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Reference graph

Works this paper leans on

18 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    Andrews, D. W. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation.Econometrica 59(3), 817–858. Andrews, D. W. (1993). Tests for parameter instability and structural change with unknown change point.Econometrica 61(4), 821–856. Andrews, D. W. and J. C. Monahan (1992). An improved heteroskedasticity and auto- correlation ...

    work page Pith review arXiv 1991

  2. [2]

    Patil, K

    Du, J.-H., P. Patil, K. Roeder, and A. K. Kuchibhotla (2023). Extrapolated cross- validation for randomized ensembles.arXiv preprint arXiv:2302.13511. Durrett, R. (2019).Probability: Theory and Examples, Volume

    work page arXiv 2023

  3. [3]

    Embrechts, P., C

    Cambridge university press. Embrechts, P., C. Kl¨ uppelberg, and T. Mikosch (2013).Modelling Extremal Events: for Insurance and Finance, Volume

    2013

  4. [4]

    Gallant, A

    Springer Science & Business Media. Gallant, A. R. (2009).Nonlinear Statistical Models. John Wiley & Sons. Hansen, B. E. (1992). Consistent covariance matrix estimation for dependent heteroge- neous processes.Econometrica 60(4), 967–972. He, Y., G. Xu, C. Wu, and W. Pan (2021). Asymptotically independent u-statistics in high-dimensional testing.Annals of S...

    2009

  5. [5]

    Kiefer, N. M. and T. J. Vogelsang (2005). A new asymptotic theory for heteroskedasticity-autocorrelation robust tests.Econometric Theory 21(6), 1130–

    2005

  6. [6]

    Kiefer, N. M., T. J. Vogelsang, and H. Bunzel (2000). Simple robust testing of regression hypotheses.Econometrica 68(3), 695–714. Korkas, K. K. and P. Fryzlewicz (2017). Multiple change-point detection for non- stationary time series using wild binary segmentation.Statistica Sinica 27(1), 287–311. Krishnan, M. (1967). The noncentral bivariate chi distribu...

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  7. [7]

    Robinson, P. M. (1991). Automatic frequency domain inference on semiparametric and nonparametric models.Econometrica 59(5), 1329–1363. Robinson, P. M. (1998). Inference-without-smoothing in the presence of nonparametric autocorrelation.Econometrica 66(5), 1163–1182. Seber, G. A. and A. J. Lee (2003).Linear Regression Analysis, Volume

    1991

  8. [8]

    hac corrections for strongly autocorrelated time series

    John Wiley & Sons. Shafer, G. and V. Vovk (2008). A tutorial on conformal prediction.Journal of Machine Learning Research 9(3), 371–421. Shao, X. (2010). A self-normalized approach to confidence interval construction in time series.Journal of the Royal Statistical Society: Series B (Statistical Method- ology) 72(3), 343–366. Shao, X. (2011a). A bootstrap-...

    2008

  9. [9]

    Empirical Size Comparison To examine the empirical size of different SS-SN statistics, we focus on the testing of multivariate mean

    A.2. Empirical Size Comparison To examine the empirical size of different SS-SN statistics, we focus on the testing of multivariate mean. Under the null, we assume the data comes from the following VAR(1) model:X t =ρI pXt−1 +ϵ t,whereϵ t iid∼N(0,Σ p). We consider four DGPs with different dependence structures: for DGP1 we letΣ p =I p; for DGP2 we let Σp ...

    2001

  10. [10]

    The Ljung-Box test gives ap-value less than 2.99×10 −12 whenp= 1, which is a strong indication that the null hypothesisH 0 :r 1 = 0 does not hold

    It appears that ˆr 1 = 0.61 and|ˆrp|<0.21 forp≥2. The Ljung-Box test gives ap-value less than 2.99×10 −12 whenp= 1, which is a strong indication that the null hypothesisH 0 :r 1 = 0 does not hold. Note that the limiting null distribution of Ljung-Box test statistic isχ 2 and is obtained under the strong iid assumption, whereas the test developed by Lobato...

    2001

  11. [11]

    To prove part (ii).3 of Theorem 2, note that √n||µn−µ0|| →0 impliesN 2, N3 and ⌊nα⌋(n−⌊nα⌋) n ||µn−µ0||2 from Equation (47) are of ordero p(1). Also from Equation (48), we have 2 ⌊nα⌋2 n ||µn−µ0||2 1 min j=1,2,...,p ˆσ2 j 1 (n−⌊nα⌋)2 nX k=⌊nα⌋+1 ||bk||2 =o p(1), soQ (M) n (α) will have the same limiting distribution as under the null hypothesis.2 Another ...

    2023

  12. [12]

    So for notation simplicity, without loss of generality, we treat {Xnt}n t=1 as{X ′ nt}n t=1 and assume the former is defined in the same probability space asW n(r). If we can show ˆjn − ˆj(W) n p →0 and max j≤pn T (D) n (α, j)−T (W) n (α, j) p →0, then the conclusion follows since for anyϵ >0, P(|T (D) n (α, ˆjn)−T (W) n (α, ˆj(W) n )|> ϵ)≤P( ˆjn ̸= ˆj(W)...

    2019

  13. [13]

    (2013), we have 1 2a[Y(pn)(a)−Y(pn−1)(a)] D→ ˜Y∼exp(1)

    By Corollary 4.2.11 in Embrechts et al. (2013), we have 1 2a[Y(pn)(a)−Y(pn−1)(a)] D→ ˜Y∼exp(1). For anyϵ >0, fix 0<˜c < a 2 such thatP( ˜Y < ˜c 2a)< ϵ C , then we have P(V (pn)−V(pn−1) <˜c)≤CP(Y (pn)(a)−Y(pn−1)(a)<˜c)→CP( ˜Y < ˜c 2a)< ϵ,(54) which implies (V(pn)−V(pn−1))−1 =O p(1) and part (b) is proved. Another Look at Bandwidth-free Inference59 Proof of...

    2013

  14. [14]

    We divide the proof into two parts: (a) 1 n n ˆP ⊤ n S⌊nα⌋+1,n o2 =N Q +o p(NQ)

    (63) If we can show 1 n ˆP ⊤ n S⌊nα⌋+1,n 2 =N Q +o p(NQ) and Z 1 α ˆP ⊤ n [χn(r)−χn(α)− r−α 1−α(χn(1)−χn(α))] 2dr=D Q +o p(DQ), then we haveQ (D) n (α)−Q (W) n (α) p →0 and the theorem is proved. We divide the proof into two parts: (a) 1 n n ˆP ⊤ n S⌊nα⌋+1,n o2 =N Q +o p(NQ). (b) R 1 α ˆP ⊤ n [χn(r)−χn(α)− r−α 1−α(χn(1)−χn(α))] 2dr=D Q +o p(DQ). Proof of ...

    2013

  15. [15]

    By simple calculation we can see that ˆh D→arg max j∈{1,2,...,p} h B(j)(b)−(B (j)(1)−B (j)(1−b)) i2 ϑ2 j = ˜h, whereB (j)(r) is thejth coordinate ofΓ 1/2Bp(r)

    Following the same argument as in the proof of Theorem 1, we haveP( ˆj̸= ˆh)→0 andP( ˜Gn ̸=G n)→0, so it suffices to prove ˜Gn D→G. By simple calculation we can see that ˆh D→arg max j∈{1,2,...,p} h B(j)(b)−(B (j)(1)−B (j)(1−b)) i2 ϑ2 j = ˜h, whereB (j)(r) is thejth coordinate ofΓ 1/2Bp(r). Also, letk=⌊τ n⌋for someτ∈ (b,1−b), by continuous mapping theorem...

    2000

  16. [16]

    Note that conditioning on{ς1, ς2, ϱ},{( ˜W1(r), ˜W2(r))⊤}r∈[0,1] d ={(ς 1B1(r), ϱς2B1(r)+ p 1−ϱ2ς2B2(r))⊤}r∈[0,1]

    Next, we show E|I (2)|=E E n P ˜W 2 1 (1) N( ˜W1) ≤a ˜B P ˜W 2 2 (1) N( ˜W2) ≤b ˜B ς1, ς2, ϱ o −P(U 1 ≤a)P(U 1 ≤b) →0. Note that conditioning on{ς1, ς2, ϱ},{( ˜W1(r), ˜W2(r))⊤}r∈[0,1] d ={(ς 1B1(r), ϱς2B1(r)+ p 1−ϱ2ς2B2(r))⊤}r∈[0,1]. Denotea(ϱ) =ϱ 2bN(B1)−ϱ2bN(B2) +bϱ p 1−ϱ2 ˜N(B1, B2) where ˜N(B1, B2) = 2 R 1 0 (B1(r)− Another Look at Bandwidth-free Infe...

    2002

  17. [17]

    +a(ϱ) =bN(ϱB 1+ p 1−ϱ2B2)>0 andbN(B 2)>0, which impliesbN(B2)+ξa(ϱ) −1 ≤bN(B2)+a(ϱ) −1 whena(ϱ)<0 and bN(B2)+ξa(ϱ) −1 ≤bN(B2) −1 whena(ϱ)>0. This implies that bN(B2)+ξa(ϱ) −1 ≤bN(B2)+a(ϱ) −1 +bN(B2) −1 , which yields E nbN(B2) +ξa(ϱ) −1 ς1, ς2, ϱ o ≤E nbN(B2) +a(ϱ) −1 ς1, ς2, ϱ o +E nbN(B2) −1 ς1, ς2, ϱ o =E nbN(ϱB1+ p 1−ϱ2B2) −1 ς1, ς2, ϱ o +C =E nbN(B1)...

    2013

  18. [18]

    The following lemma comes from Devroye et al

    be the Borelσ-algebra onR 2 and forR 2-valued random vectorsX,Y, let||X− Y||T V = sup A∈B(R2) |P(X∈A)−P(Y∈A)|be the total variation distance. The following lemma comes from Devroye et al. (2018). Lemma 8.For any bivariate normal random vectorsX∼ N(0,Σ 1),Y∼ N(0,Σ 2)where bothΣ 1 andΣ 2 are positive definite, we have ||X−Y|| T V ≤3||Σ −1/2 2 Σ1Σ−1/2 2 −I 2...

    2018