An analytical random-rotation test for autocorrelation showed its clearest advantage at selected higher lags, according to a new arXiv preprint. In the reported higher-lag AR experiment at lag 3, the post-beta rotation test had the strongest power while maintaining the reported correct empirical test size. In a separate multi-lag simulation, Fisher's combination of rotation p-values had superior power to all other tests considered. The gains were not uniform: Durbin-Watson led at lag 1, and single-lag rotation tests were weaker than omnibus tests for MA(1) and MA(2).
A formula built for selected lags
The proposal centers on a beta-corrected statistic presented as a closed-form expression, avoiding numerical inversion. The framework is also presented as extending to permutation and reflection groups and to arbitrary symmetric quadratic forms.
Simulation results were mixed
Before comparing detection power, the authors tested calibration under null conditions. The null-calibration experiment generated 5,000 time-series replications of length 1,000 under iid t(4) and normal-noise models. Empirical size—the observed rejection rate under the null—was reported as correct for Gaussian and moderately heavy-tailed noise. Under t(2) noise, an infinite-variance setting, the rotation test produced conservative p-values.
An AR(1) power experiment simulated 2,000 time-series datasets of length 1,000 for each coefficient from 0 through 0.15, with normal noise or t noise with 1, 2 or 4 degrees of freedom. Here, power means the ability to detect the autocorrelation built into the simulated series. At lag 1, Durbin-Watson had the strongest power; the rotation test matched Breusch-Godfrey and Ljung-Box, while the uncorrected concentration approach had no statistical power.
In the higher-lag AR(h) experiment illustrated at lag 3, the post-beta rotation test had the strongest power while maintaining correct empirical test size. The multi-lag experiment used autoregressive terms at lags 2 and 4, coefficient values from 0 to 0.15, 2,000 series of length 1,000, and normal or t(2) noise. Combining rotation p-values for selected lags with Fisher's method was reported to yield superior power to every other considered test.
The results changed for moving-average models. For MA(1) and MA(2) processes, single-lag rotation tests had lower power than omnibus tests; Fisher combination matched comparator power for MA(1), with similar behavior for MA(2). The computation comparison used samples of 100 observations over 3,000 repetitions, and each group-based p-value was estimated from 100 random transformations. Brute-force randomization methods had similar power in the reported comparison, while the analytic concentration test lost only a little power and was computationally trivial.
A solar series gives a more specific signal
The paper's observational test used monthly solar intensity from January 1900 through February 2025, a series of 1,502 months. It used three smoothing bandwidths: 6, 12 and 18 months. In the residual analysis, rotation testing found significant autocorrelation at lags 1, 6, 7, 8 and 9 at the 0.1% level. After an AR(1) fit, only lags 6 and 9 were identified at that level.
At long lags, the correction for multiple testing produced different figures depending on the smoothing choice. After Benjamini-Hochberg correction, lag 135 months had reported false-discovery rates of 0.012, 0.013 and 0.06 for the 6-, 12- and 18-month bandwidths, respectively.
Evidence remains bounded
Those results are best read as evidence about the tested settings, not as a universal ranking of autocorrelation tests. The work is a version-1 arXiv preprint, and the supplied analysis covers mathematical derivations, specified AR and MA simulations, and one observational solar-intensity series. The calibration warning is also clear: t(2) noise with infinite variance produced conservative rotation p-values. The solar long-lag result varied from 0.012 to 0.06 across the three smoothing bandwidths.
The authors report funding support from the Natural Sciences and Engineering Research Council of Canada.
Paper data and sources
Original title: Random Invariance Testing on Quadratic Form Statistics with Application to Autocorrelation
Authors: Amitakshar Biswas, Adam B Kashlak
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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