Cuscore Statistic: Theory & Applications

• Cuscore statistic is a sequential test tool that employs an L^p-functional of a weighted CUSUM process to detect shifts in mean or parameters in time series data.
• It leverages rigorous asymptotic properties and explicit threshold formulas to balance sensitivity and specificity in changepoint detection.
• Efficient implementations, such as the FOCuS algorithm, enable real-time, online monitoring with dynamic updating and low computational complexity.

The Cuscore statistic is a class of sequential test statistics for detecting changes in the mean or other parametric shifts in time series and experimental measurement data. Foundationally connected to the established CUSUM family, the Cuscore is defined as an $L^p$-functional of a weighted cumulative-sum (CUSUM) process, generalizing both the classic CUSUM and likelihood-based sequential detectors. This approach provides rigorous asymptotic properties for threshold selection and supports a spectrum of practical changepoint detection and online monitoring schemes (Horváth et al., 2020, Romano et al., 2021, Zhang et al., 30 Dec 2025).

1. Mathematical Construction

Let $\{X_1, \dots, X_N\}$ be observations modeled as $X_i = \mu+\epsilon_i$, with $\mu$ constant under the null hypothesis $H_0$ and $\{\epsilon_i\}$ a (possibly weakly dependent) noise sequence fulfilling a strong approximation to Brownian motion (Horváth et al., 2020). The unweighted CUSUM process is

$$Z(x) = \sum_{i=1}^{\lfloor x \rfloor} X_i - (\lfloor x \rfloor/N) \sum_{i=1}^N X_i,\quad 0\leq x\leq N,$$

with time-scaled variant

$Z_N(t) = N^{-1/2} Z((N+1)t),\quad 0<t<1.$

Introducing a continuous weight $w(t)>0$ (bounded away from zero on compact subsets), the $L^p$-Cuscore statistic is

$T_{N,p}(w) = \int_0^1 \frac{|Z_N(t)|^p}{w(t)}\,dt.$

For $p=1$ and constant $w$, this reduces to the classical CUSUM; for higher $p$ and general $w(t)$, it enables sensitivity to distributed change-patterns and weighted emphasis on particular time regions (Horváth et al., 2020).

An alternative construction adapted for parametric change detection, especially in sequential experiments, is as follows. With observed data

$y_t = T + a_t + \theta x_t,$

where $T$ is an unknown baseline, $a_t \sim \mathcal{N}(0,\sigma_a^2)$, and $x_t$ is a known function, the Cuscore is

$$Q_t(\theta_0) = \sum_{i=1}^t [y_i - T - \theta_0 x_i] x_i,$$

which recovers the classic CUSUM as a special case for $x_i=1$ (Zhang et al., 30 Dec 2025).

2. Asymptotic Theory and Limit Results

Under the null hypothesis $H_0$, and technical assumptions (functional invariance, bounded weight, integrability), the weighted $L^p$-Cuscore $T_{N,p}(w)$ admits a functional limit theorem: $$T_{N,p}(w) = \int_0^1 \frac{|Z_N(t)|^p}{w(t)}\,dt \;\Rightarrow\; T_p := \int_0^1 \frac{|\sigma B(t)|^p}{w(t)}\,dt,$$ where $B(t)$ is a standard Brownian bridge and $\sigma^2$ the innovation variance (Horváth et al., 2020). For standardized weights $w(t) = (t(1-t))^{1+p/2}$, $T_{N,p}(w)$ grows logarithmically, and after explicit centering and scaling becomes asymptotically normal: $$\frac{T_{N,p}(w)-2b(p)\ln N}{\sqrt{4a(p)\ln N}} \Rightarrow N(0,1),$$ where $a(p), b(p)$ are explicit functionals of the standard Gaussian.

For heavier weights and truncated intervals, the rescaled statistic converges to a deterministic sum of functionals of independent Wiener processes, yielding easily tabulated threshold constants without need for heavy simulations.

These limit results provide an explicit foundation for threshold selection in sequential change detection at a prescribed significance level $\alpha$.

3. Sequential Testing and Threshold Choice

The Cuscore and its centered version admit a rigorous sequential test framework, notably when distinguishing between $H_0: \theta=\theta_0$ and $H_1: \theta=\theta_1$ in parametric settings: $$Q_t(\bar\theta) > \frac{\sigma_a^2 \ln(1/\alpha)}{\theta_1-\theta_0}, \quad \bar\theta = \tfrac12(\theta_0+\theta_1),$$ with two-sided versions accumulating positive and negative deviations, and separate thresholds $h^+, h^-$ for each direction (Zhang et al., 30 Dec 2025).

For the $L^p$-functional case, thresholds are either determined by simulation of the limiting Brownian bridge integral or, in the log-divergent regime, by explicit analytic centering/scaling formulas. This yields transparent, asymptotically valid $\alpha$-level tests, removing the need for empirical or ad hoc threshold selection (Horváth et al., 2020).

4. Online and Functional Implementation

The Cuscore statistic underpins efficient online algorithms such as the FOCuS (Functional Online CuSUM) scheme. FOCuS computes a functional CUSUM statistic across all candidate post-change means $\mu$ simultaneously: $$Q_n(\mu) = \max_{0\leq s < n} \sum_{t=s+1}^n \mu \left(x_t - \frac{\mu}{2}\right),$$ with maximization over $\mu$ yielding the classical max-over-window form (MOSUM/Page-CUSUM equivalence). The algorithm maintains a dynamically pruned set of piecewise-quadratic summaries, achieving amortized $O(1)$ update and $O(\log n)$ maximization per observation, with provably tight complexity bounds (Romano et al., 2021).

A similar structure applies to the real-time Cuscore implementation for detector monitoring, where an exponentially weighted moving average (EWMA) of the baseline is recursively updated for robustness to drifts, with each update and test of $O(1)$ complexity (Zhang et al., 30 Dec 2025).

5. Practical Tuning and Implementation

Selection of operational parameters (bin size, significance level $\alpha$, reference state difference $|\theta_1-\theta_0|$, and discount $\lambda$ for EWMA) is guided by analytic variance considerations and detector resolution. False-alarm rates are explicitly controlled via the threshold formula, balancing sensitivity and specificity. Typical implementations use bin sizes large enough for normal approximation ($\geq 50$–$100$), $\alpha$ in the range $10^{-3}$–$10^{-2}$, $|\theta_1-\theta_0|$ as a function of intrinsic detector resolution or noise scale, and EWMA $\lambda \geq 0.99$ for long-term monitoring (Zhang et al., 30 Dec 2025).

6. Applications and Empirical Results

The Cuscore and Centred Cuscore framework have been employed in state-change detection in nuclear physics experiments, such as in charge-changing reaction studies at the GSI FRS and the IMP RIBLL2 beamline. In these cases, offline analysis segmented data using the Cuscore, improving parameter estimates (e.g., cross-section by $\sim2$ mb), and online EWMA-Cuscore monitoring enabled real-time detection of detector anomalies, such as beam interruptions from foreign obstructions (Zhang et al., 30 Dec 2025). The FOCuS algorithm demonstrates state-of-the-art online changepoint detection with rigorous complexity guarantees and robustness, suitable for high-frequency server data streams (Romano et al., 2021).

7. Interpretation and Comparative Advantages

The $L^p$ Cuscore, with a bounded or standardized weight, offers sensitivity advantages over the supremum-norm CUSUM—faster $\sqrt{N}$-rate convergence and increased power for distributed rather than localized changes. The standardized weight leads to simple asymptotic normality and explicit thresholds, while heavy/truncated weighting isolates changes at boundaries or other regions of interest.

Key theoretical developments by Horváth and Rice (2020) establish null limit laws, ensuring theoretically justified critical values in a wide range of mean-stationarity test scenarios. The generality, efficiency, and ease of threshold computation position the Cuscore as a central method in modern changepoint detection research (Horváth et al., 2020, Romano et al., 2021, Zhang et al., 30 Dec 2025).