Self-Normalized Mean Change Detection

• The paper introduces self-normalized test statistics that detect mean changes without explicit nuisance parameter estimation, ensuring robust inference.
• The methodology extends to various data types including independent, dependent, high-dimensional, functional, and locally stationary processes.
• The approach supports multiple change-point localization using recursive segmentation and simulation-calibrated thresholds for finite-sample accuracy.

Self-normalized extensions for mean changes constitute a class of inferential methodologies for detecting changes in the mean of sequential data, built around ratios of test statistics and variance estimators constructed from the data itself. These approaches enable strong theoretical guarantees and robust finite-sample performance, even in the absence of explicit estimation of nuisance parameters such as the long-run variance. The self-normalization framework has been systematically developed for independent and dependent univariate data, long-range dependent and locally stationary sequences, high-dimensional and functional data, and forms the backbone of multiple-segmentation algorithms in complex scenarios.

1. Foundational Self-Normalized Mean Change Tests

The self-normalized change-in-mean test introduced by Csörgő–Hut (Csorgo et al., 2013) provided a rigorous framework for detecting an at-most-one change in the mean of a sequence of independent observations $X_1, \ldots, X_n$ with a common but unknown mean. The core construction is the maximal self-normalized deviation statistic,

$$T_n = \max_{2\leq k\leq n-2} \frac{ | S_k - \frac{k}{n} S_n | }{ \sqrt{ \frac{k(n-k)}{n} }\, V_n(k) },$$

where $S_k$ is the cumulative sum up to time $k$, and $V_n(k)$ is the within-segment pooled sample standard deviation,

$V_n(k) = \sqrt{ U_{k,n}^2 + V_{k,n}^2 }.$

Here, $U_{k,n}^2$ and $V_{k,n}^2$ are unbiased sample variances before and after time $k$. Under the null hypothesis (no change in mean), and under a finite $\mathbb{E} X^2 \log\log(|X|+1) < \infty$ moment condition, $T_n$ (suitably normalized) converges weakly to a standard Gumbel law: $$\Pr(a_n T_n - b_n \leq x) \rightarrow \exp(-e^{-x}),$$ with $a_n = \sqrt{2\log\log n}$ and $$b_n = 2\log\log n + \frac12\log\log\log n - \frac12\log\pi$$. The same limit holds for infinite variance $X_i$ in the domain of attraction of the normal law under regular variation conditions.

In the presence of a single mean change, $T_n$ diverges in probability, implying strong consistency for any diverging rejection threshold $c_n$ with $c_n = o(b_n/a_n)$ (Csorgo et al., 2013).

2. Extensions to Dependent, Long-Range Dependent, and Locally Stationary Time Series

The classical self-normalized approach is well suited to independent or weakly dependent data. For strong dependence or local stationarity, modifications are necessary.

• Long-Range Dependence: For Gaussian subordinated long-range dependent processes, self-normalized Wilcoxon-type tests are constructed via partial sums of ranks and suitably self-normalized statistics (Betken, 2014). This allows detection of mean changes where the serial dependence precludes straightforward estimation of the variance. The critical value is determined via simulations of Hermite processes (fractional Brownian motion if Hermite rank $=1$).
• Locally Stationary Processes: In environments where the variance function $\sigma^2(t)$ varies over time, classical factorization of the long-run variance fails (Heinrichs, 8 Sep 2025). Heinrichs developed a bivariate CUSUM-based approach relying on partial-sum arrays $S_n(t,s)$ over blocks permuted sequentially, and constructs a test statistic $T_n$ as a ratio of two suprema over different marginals of the process:

$T_n = \frac{ \sup_s |V_n(s)| }{ \sup_s |H_n(s)| }.$

This approach yields a pivotal limit law determined by suprema of independent Brownian motions, ensuring correct type I error control and power against broad alternatives, without the need for direct variance estimation (Heinrichs, 8 Sep 2025).

3. High Dimensional and Functional Extensions

• High-Dimensional Data: For $\{Y_t\}_{t=1}^n \in \mathbb{R}^p$ with $p \to \infty$, self-normalized tests are based on U-statistics of the Chen–Qin form for mean change, with normalization constructed from the sum of squared projected contrasts (Wang et al., 2019). Tests are formulated as maximizations over all cut points. In the dependent case, trimming is applied to reduce edge effects, and Monte Carlo calibration produces thresholds. These methods retain pivotality under the null and strong power for dense alternative settings. The wild binary segmentation wrapper allows multiple change-point localization.
• Functional Data: For strictly stationary $L^2$-valued time series, relevant mean changes (i.e., $L^2$-distance between means exceeding $\Delta$) are detected using self-normalized test statistics of the form $$W_n = (\hat{\mathbb{T}}_n - \Delta)/\hat{\mathbb{V}}_n$$, where $\hat{\mathbb{T}}_n$ is the empirical squared mean and $\hat{\mathbb{V}}_n$ is a self-normalizer integrating deviations of partial-sum curves over a grid. The resulting limit law is pivotal and critical values are obtained via simulation (Dette et al., 2018).

4. Multiple Change-Point Algorithms and Segmentation

To deal with multiple changes, self-normalized statistics are embedded in recursive segmentation algorithms:

• SNCP Algorithm: The nested local-window self-normalization framework scans overlapping neighborhoods of candidate change-points and computes local statistics to obtain consistent estimates of both the number and locations of changes, regardless of dependence structure. For a segment $[a, b]$, the maximum self-normalized CUSUM statistic is computed over all sufficiently sized local windows, and binary segmentation is applied recursively (Zhao et al., 2021).
• Wild Binary Segmentation (WBS): High-dimensional SN change-point detection is extended via WBS, where self-normalized statistics are evaluated on random subintervals to locate breaks adaptively (Wang et al., 2019). Pivotal null distributions remain available via simulation.

Model/Data Type	Statistic Structure	Key Reference
i.i.d. Univariate	Max self-normalized deviation over $k$	(Csorgo et al., 2013)
Long-range Dependent	Max self-normalized rank/Wilcoxon	(Betken, 2014)
Locally Stationary	Bivariate CUSUM ratio statistic	(Heinrichs, 8 Sep 2025)
High-dimensional	U-statistics, trimmed SN norm	(Wang et al., 2019)
Functional	$\|\mu\|_{L^2}$ SN (partial-sum based)	(Dette et al., 2018)
Segmentation/Multiple breaks	Local-window/recursive SN CUSUM	(Zhao et al., 2021)

5. Asymptotic Theory and Distributional Limits

The common feature across these methodologies is that the normalization is data-driven and constructed to asymptotically cancel unknown nuisance parameters (long-run variance, unknown scale), resulting in pivotal limit distributions under the null hypothesis:

• Independent Case: Gumbel-type limits, as in the classical Darling–Erdős theorems for maxima of normalized partial sums (Csorgo et al., 2013).
• Dependent/Functional/High-Dimensional: Functionals (suprema, ratios) of Brownian motion, bridges, Hermite processes, and Gaussian random fields, often tabulated via simulation for finite-sample inference (Betken, 2014, Heinrichs, 8 Sep 2025, Dette et al., 2018, Wang et al., 2019, Zhao et al., 2021).

Under alternatives (single or multiple changes), numerator effects diverge in probability while the normalizers remain $O_P(1)$, yielding consistency. Local alternatives, where the mean shift shrinks with $n$, produce non-central limit processes and allow for explicit characterization of power properties.

6. Implementation, Calibration, and Finite-Sample Considerations

Self-normalized mean change tests lend themselves to tuning-free or simulation-based calibration:

• No estimation of long-run variance or dependence parameters is needed.
• All critical values are obtained via Monte Carlo simulation of the limiting processes (Brownian motion, Hermite processes, etc.), using sample-size-appropriate configurations (Betken, 2014, Heinrichs, 8 Sep 2025).
• Trimming or windowing parameters in dependent or high-dimensional cases are set either by theory (e.g., exclusion of very early/late break points) or simple heuristics (e.g., fixed fractions of the sample).
• Implementation overhead is moderate and scales linearly in $n$ in most cases; for high-dimensional or massively functional data, computational cost is dominated by matrix operations or repeated partial-sums.

Simulation studies consistently confirm type I error control, strong power for moderate-to-large changes, and robustness to serial dependence, heavy tails, and heteroscedasticity (Betken, 2014, Heinrichs, 8 Sep 2025, Zhao et al., 2021, Dette et al., 2018).

7. Extensions, Limitations, and Future Directions

Self-normalized testing provides a model-agnostic, theoretically grounded approach to mean change inference. Ongoing and potential extensions include:

• Broader statistics: variance, quantile, correlation change detection (Zhao et al., 2021).
• Multivariate, functional, and high-dimensional segmentation (Wang et al., 2019, Zhao et al., 2021, Dette et al., 2018).
• Joint detection for covariance structure changes (Wang et al., 2019).
• Adaptive or data-driven trimming/localization in highly heterogeneous environments (Betken, 2014, Zhao et al., 2021).
• Automated and robust segmentation of complex mean structure via local-window or wild binary segmentation algorithms (Wang et al., 2019, Zhao et al., 2021).
• Limitation: for small sample sizes or multiple closely spaced change points, care is required in setting window and trimming parameters to avoid boundary effects and loss of local power.

Self-normalized extensions for mean changes have proven theoretically optimal and practically competitive across statistical change-point inference regimes, providing pivotal inference for complex data scenarios with minimal assumptions on the error process or dimensionality.