Statistical Inference for High-Dimensional Random Processes

Statistical inference for high-dimensional random processes addresses the problem of making reliable conclusions when the number of variables or dimensions grows as fast as, or faster than, the sample size. Traditional asymptotic tools break down in such regimes, prompting new theory for central limit theorems and non-asymptotic concentration inequalities where dimension and sample size co-evolve. Modern methods employ resampling techniques, moment-adaptive procedures and tailored test statistics to control error rates and quantify uncertainty in settings ranging from genomics and finance to climate modelling and network analysis. Core challenges include dependence across time or space, heavy-tailed distributions and computational tractability. Recent progress has focused on sharpened lower-tail bounds for Gaussian maxima, refined bootstrap consistency under weak dependence, and novel goodness-of-fit criteria that remain valid as dimensionality grows. These advances collectively mitigate the curse of dimensionality, enabling robust hypothesis testing, confidence set construction and change-point detection in complex stochastic systems.

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High-dimensional bootstrap methods have been extended to address simultaneous inference across many parameters. A comprehensive review of bootstrap consistency results establishes high-dimensional central limit theorems for sample mean vectors in arbitrary rectangular regions, and demonstrates applications such as the construction of simultaneous confidence sets, step-down multiple hypothesis testing and post-selection inference. These results also underpin methods for policy-evaluation in econometrics, highlighting the broad impact of bootstrapping in modern data-rich environments.

For dependent data, a blockwise bootstrap two-sample test has been proposed for high-dimensional time series. This approach derives high-dimensional central limit theorems for α-mixing sequences under both bounded-moment and exponential-tail conditions. By resampling contiguous blocks rather than individual observations, the procedure preserves temporal dependence and yields valid critical values without assuming sample independence. Numerical studies demonstrate its power in detecting change points and distributional shifts in multivariate time series.

A novel Cramér–von Mises-type test has been developed for assessing goodness of fit in high-dimensional continuous distributions. The test statistic is based on quadratic functionals of empirical stochastic processes, and two calibration strategies are introduced: a plug-in method that uses estimated parameters directly, and a subsampling technique that constructs an empirical distribution of the statistic through repeated partial resampling. Both approaches deliver accurate size control and good power in simulations, providing practitioners with flexible tools for distributional assessment in large dimensions.

Statistical Inference for High-Dimensional Random Processes publication trend

The graph below shows the total number of articles in statistical inference for high-dimensional random processes across all publications each year (not limited to Nature Index journals).

Technical terms

High-dimensional central limit theorem: Extension of the classical central limit theorem to settings where the parameter dimension grows with sample size.

Bootstrap: Resampling method that approximates the sampling distribution of a statistic by repeatedly drawing samples with replacement from the observed data.

α-mixing sequence: A type of dependent sequence whose correlations decay at a specified rate, ensuring weak dependence over time.

Blockwise bootstrap: A resampling technique for time-series data that draws contiguous blocks to preserve dependence structure.

ℓ∞-type test statistic: A maximum-norm statistic that evaluates the largest absolute deviation across multiple dimensions.

Cramér–von Mises test: A goodness-of-fit test based on the integrated squared difference between empirical and theoretical distribution functions.

Subsampling: Method of repeatedly computing the statistic on smaller subsets of data to approximate its distribution without replacement.

References

1. High-Dimensional Data Bootstrap. Annual Review of Statistics and Its Application (2023).
2. A Blockwise Bootstrap-Based Two-Sample Test for High-Dimensional Time Series. Entropy (2024).
3. A High-Dimensional Cramér–von Mises Test. Mathematics (2024).
4. A sharp lower-tail bound for Gaussian maxima with application to bootstrap methods in high dimensions. Electronic Journal of Statistics (2022).