Window-Limited GLR-CuSum Detection

• Window-Limited GLR-CuSum is a sequential change detection method that uses a sliding window to estimate unknown post-change distributions, ensuring fast adaptation.
• It achieves first-order asymptotic optimality by balancing statistical precision and computational efficiency through window-restricted likelihood maximization.
• Practical implementations leverage recursive data structures and window size tuning to maintain low per-sample cost in both parametric and nonparametric settings.

The window-limited generalized likelihood-ratio cumulative sum (GLR-CuSum) procedure is a family of sequential change detection methods designed for quickest change detection (QCD) when the post-change distribution is unknown and inferred online using only a finite sliding window of recent observations. These procedures were introduced to reconcile statistical optimality, computational tractability, fast adaptation, and robustness in both parametric and nonparametric settings. Window-limited GLR-CuSum achieves first-order asymptotic optimality under broad model assumptions, and its recent generalizations extend to nonparametric, nuisance, and nonstationary post-change regimes (Liang et al., 2023, Lau et al., 2019, Xie et al., 2022, Liang et al., 2021, Lau et al., 2019).

1. Quickest Change Detection and the Need for Window Limitation

In the classical QCD setup, observations $X_1,X_2,...$ are independent; for an unknown change-point $\nu$, the pre-change regime ($n<\nu$) has distribution $f_0$ (known), and the post-change regime ($n\geq\nu$) has distribution $f_1$ (typically unknown or partially specified). The objective is to minimize the worst-case detection delay

$\WADD(T)=\sup_{\nu\ge1}\esssup\,\E_\nu\bigl[(T-\nu+1)^+\mid\F_{\nu-1}\bigr]$

subject to a false-alarm constraint

$\E_\infty[T]\ge\gamma$

for a (large) target $\gamma$. While CuSum and likelihood-ratio-based schemes achieve minimax optimality when $f_1$ is known, in practice $\nu$0 is often unknown and must be estimated. Standard GLR-CuSum statistics scan back over the entire observed history at each time, causing both computational and statistical delays (e.g., large burn-in). Window-limited GLR-CuSum restricts estimation and maximization to the most recent $\nu$1 samples, maintaining optimality while drastically improving efficiency and responsiveness (Liang et al., 2023, Lau et al., 2019, Liang et al., 2021, Xie et al., 2022).

2. Window-Limited GLR-CuSum: Definitions and Algorithm

The generic window-limited GLR-CuSum statistic at time $\nu$2 takes the form: $\nu$3 where $\nu$4 is the set of candidate change-points inside an $\nu$5-length sliding window, and $\nu$6 denotes the post-change density parameterized by $\nu$7 (possibly nonstationary, i.e., dependent on time since the putative change at $\nu$8). For a fixed or suitably growing window size $\nu$9, the stopping rule is

$n<\nu$0

where threshold $n<\nu$1 is chosen so that $n<\nu$2 for the desired average run length (ARL) constraint (Liang et al., 2021, Xie et al., 2022).

Nonparametric variants (NGLR-CuSum) replace parametric likelihoods with sliding-window density estimators (e.g., kernel density estimates computed on samples $n<\nu$3 for leave-one-out correction), and use

$n<\nu$4

with the statistic

$n<\nu$5

and otherwise identical stopping rule (Liang et al., 2023).

3. Statistical Properties and Optimality

Window-limited GLR-CuSum schemes are proven to be first-order asymptotically optimal in Lorden's sense: $n<\nu$6 where $n<\nu$7 is a generalized Kullback–Leibler information number dependent on the post-change regime. For parametric and certain nonstationary models, the cumulative or instantaneous mean KL-divergence governs the detection-delay scaling (Liang et al., 2021, Xie et al., 2022). For nonparametric density estimation, rates depend on estimator convergence: $n<\nu$8 ensuring optimal WADD as $n<\nu$9 if the window $f_0$0 grows at least linearly in $f_0$1 (threshold) and $f_0$2 (Liang et al., 2023).

False-alarm rates are controlled via renewal-theoretic calculations, motivating thresholds of the form $f_0$3 with higher-order corrections for nonparametric estimators (to account for "max-product" bounds) (Liang et al., 2023).

Under suitable conditions, window-limited GLR-CuSum procedures match the minimax lower bounds for detection delay while guaranteeing ARL constraints—robust to misspecification and model complexity (Liang et al., 2021, Xie et al., 2022, Liang et al., 2023).

4. Window Size Selection and Practical Considerations

Asymptotic results require the window size $f_0$4 to satisfy: $f_0$5 where $f_0$6 is the inverse growth of the cumulative post-change KL divergence, capturing the typical delay scale. In i.i.d. or stationary settings, $f_0$7 suffices; nonparametric tests employ $f_0$8 or $f_0$9 for $n\geq\nu$0 (Liang et al., 2023, Liang et al., 2021). Window sizes that grow too slowly result in suboptimal delay, while excessive growth incurs unnecessary computation. For practical tuning, one sets $n\geq\nu$1 close to $n\geq\nu$2 and verifies estimator-specific conditions (e.g., the "max-product" moment, leave-one-out cross-validation) (Liang et al., 2023).

Efficient recursive implementations exploit monotone-queue data structures and partial sum representations to keep per-sample cost $n\geq\nu$3 for classical and $n\geq\nu$4 for composite parameter or nonparametric settings (where $n\geq\nu$5 is the number of grid points for parameter discretization) (Liang et al., 2021, Xie et al., 2022, Liang et al., 2023).

5. Model Variants and Extensions

Window-limited GLR-CuSum accommodates a broad array of QCD problems, including:

• Nonparametric uncertainty: NGLR-CuSum achieves minimax rates without any prior knowledge of $n\geq\nu$6, driven solely by the smoothness (e.g., Hölder class) and convergence properties of the density estimator (Liang et al., 2023).
• Nonstationary post-change: Allows $n\geq\nu$7 to vary with time since change. Asymptotic delay is governed by a function $n\geq\nu$8 tracking cumulative expected post-change divergence; minimax-optimality holds under regularity and concentration-of-measure assumptions (Liang et al., 2021).
• Nuisance changes: The window-limited GLR form extends to discriminating critical changes from nuisance shifts by maximizing test statistics over all candidates for the nuisance-change point, retaining optimality provided identifiability via KL-separation (Lau et al., 2019).
• Sampling constraints and switching costs: The methodology extends to settings with restricted observation policies, capturing both observation sparsity and switching penalizations via randomized finite-window patterns and Markovian policy synthesis (Lau et al., 2019).

6. Numerical Performance and Empirical Validation

Extensive numerical experiments validate theory across several canonical and applied settings:

• Gaussian mean shift (unknown mean): NGLR-CuSum and parametric window-limited GLR-CuSum achieve detection delays indistinguishable from the oracle CuSum with known mean, as long as window sizes are chosen per the theoretical guidelines. For moderate $n\geq\nu$9, ARL and delay scale as predicted by renewal theory and minimax optimality (Liang et al., 2023, Xie et al., 2022).
• Exponential and decaying mean models: In scenarios with nonstationary or composite post-change parameters, window-limited GLR-CuSum tracks optimal known-parameter detection performance for large enough $f_1$0; the influence of window selection is especially pronounced for heavy-tailed or fast-varying post-change models (Liang et al., 2021).
• Nuisance regimes: In the presence of nuisance changes, window-limited GLR-CuSum outperforms moving-average and two-stage procedures in both theoretical and empirical comparisons (Lau et al., 2019).
• Applied contexts (e.g., pandemic monitoring): Real-world data such as COVID-19 incidence timeseries illustrate the practical effectiveness of window-limited GLR-CuSum in rapidly and reliably identifying change points corresponding to major trend shifts (Liang et al., 2021).

7. Relation to Classical and Contemporary QCD Schemes

The window-limited GLR-CuSum generalizes several classical QCD detectors:

Method	Post-change knowledge	Statistic	Windowed?	Asymptotic optimality
CuSum	$f_1$1 known	LLR sum	No	Yes
Parametric GLR-CuSum	$f_1$2 in known family	Sup over $f_1$3	Optional	Yes, windowed or not
Window-limited GLR-CuSum	$f_1$4 in (rich) family	Windowed sup	Yes	Yes
NGLR-CuSum	$f_1$5 unknown, smooth	Nonparametric LR	Yes	Yes with estimator

These approaches constitute a unified toolkit for detection under uncertainty, offering trade-offs between computational load, adaptation lag, and distributional assumptions. For many realistic applications, windowing is both necessary and sufficient for minimax-optimal QCD (Liang et al., 2023, Lau et al., 2019, Xie et al., 2022, Liang et al., 2021, Lau et al., 2019).