Time-Varying Thresholds in CUSUM Change Detection
Summary
The document asks whether a symmetric CUSUM filter can use a predictable threshold that changes with estimated volatility, such as an EWMA or rolling estimate. The motivation is to make event triggers respond to moves relative to the current volatility regime rather than a fixed absolute move. It distinguishes this theoretical question from whether the approach works empirically.
No answer or evidence is provided; the document frames issues that an analysis would need to address. A changing threshold may be a coherent adaptive variant, but classical fixed-threshold properties cannot simply be assumed to carry over. The threshold series can affect how observations accumulate and when events occur, while estimated volatility may add dependence on past data. False-alarm behavior and interpretation therefore require separate analysis, and the document notes that related adaptive methods may modify more than the threshold alone.
Key ideas
- The document considers replacing CUSUM’s fixed threshold with a predictable, volatility-based threshold.
- The motivation is to make triggers reflect moves relative to the prevailing volatility regime.
- It asks whether this preserves the interpretation and guarantees of classical fixed-threshold CUSUM.
- Threshold adaptation may alter path dependence and false-alarm properties, which require analysis.
- The document raises the question but supplies no theoretical resolution or empirical results.
Tags
Full text
# Dynamic threshold in CUSUM # Dynamic threshold in CUSUM The symmetric CUSUM filter, as presented in Advances in Financial Machine Learning, $\S$2.5.2.1, and implemented in mlfinpy, takes a time series and a scalar threshold $h$, and triggers an event when the cumulative positive or negative deviation exceeds that threshold. In financial data, volatility is clearly time-varying, so I am wondering whether it is theoretically reasonable to replace the scalar threshold $h$ with a predictable time-varying threshold $h_t$, for example one based on an EWMA or rolling estimate of volatility. Intuition: in quiet periods I would want a lower threshold, and in volatile periods a higher threshold, so that the filter reacts to moves relative to the current regime rather than to a fixed absolute scale. My question is not whether this is useful empirically, but whether it is theoretically coherent: - Does using a time-varying threshold $h_t$ still fit within the logic of CUSUM? Or does it break the standard interpretation / guarantees of the classical fixed-threshold CUSUM? - Is this best thought of as a legitimate volatility-normalized variant of CUSUM, or as a different procedure entirely? - Are there specific pitfalls, such as added path dependence or altered false-alarm properties, that limit the practical use of this naive substitution $h \to h_t$? I found related work on adaptive CUSUM methods, but those papers seem to modify more than just the threshold: - Ahad, Davenport, Xie, Data-Adaptive Symmetric CUSUM for Sequential Change Detection (2024/2025 availability), https://doi.org/10.1080/07474946.2023.2272908 - Yi and Qiu, An adaptive CUSUM chart for drift detection (2022), https://doi.org/10.1002/qre.3020 Those seem more sophisticated than simply feeding a threshold series into a standard CUSUM implementation. I would appreciate a theoretical answer in plain terms: what exactly is lost, preserved, or changed when the threshold becomes time-varying?
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