A New Class of Weighted CUSUM Statistics

Area of Science:

• Statistics
• Data Analysis
• Time Series Analysis

Background:

• Change point detection is crucial for identifying shifts in data patterns.
• Existing cumulative sum (CUSUM) statistics may not fully utilize prior knowledge of change point locations.
• Prior information, such as change points in sequence tails, requires specific statistical methods.

Purpose of the Study:

• To develop a new class of weighted CUSUM statistics that incorporate prior information on change point locations.
• To provide a theoretical framework for these weighted CUSUM statistics, including exact and asymptotic distributions.
• To extend the application of these methods to various statistical models and real-world data.

Main Methods:

• Proposed weighted cumulative sum (CUSUM) statistics with quadratic weighting schemes.
• Derived exact distributions using eigenvalues for normal models.
• Investigated asymptotic distributions and compared them with established statistics.
• Extended methodologies to graphical models, mixture of normals, Poisson, and weakly dependent data.

Main Results:

• The proposed weighted CUSUM statistics explicitly incorporate prior knowledge of change point positions.
• Exact and asymptotic distributions were derived, offering valuable theoretical insights.
• The new statistics showed improved power over graph-based methods in simulations.
• Demonstrated applicability to complex models and video data detection problems.

Conclusions:

• Weighted CUSUM statistics offer a flexible and powerful approach for change point detection when prior location information is available.
• The theoretical framework provides a solid foundation for statistical inference.
• The method is adaptable to diverse data types and complex statistical models, enhancing detection capabilities.