Sequential monitoring of a Bernoulli sequence when the pre-change parameter is unknown

The task of monitoring for a change in the mean of a sequence of Bernoulli random variables has been widely studied. However most existing approaches make at least one of the following assumptions, which may be violated in many real-world situations: (1) the pre-change value of the Bernoulli parameter is known in advance, (2) computational efficiency is not paramount, and (3) enough observations occur between change points to allow asymptotic approximations to be used. We develop a novel change detection method based on Fisher’s exact test which does not make any of these assumptions. We show that our method can be implemented in a computationally efficient manner, and is hence suited to sequential monitoring where new observations are constantly being received over time. We assess our method’s performance empirically via using simulated data, and find that it is comparable to the optimal CUSUM scheme which assumes both pre- and post-change values of the parameter to be known.

[1]  Michèle Basseville,et al.  Detection of abrupt changes: theory and application , 1993 .

[2]  Changliang Zou,et al.  Nonparametric control chart based on change-point model , 2009 .

[3]  Douglas C. Montgomery,et al.  Introduction to Statistical Quality Control , 1986 .

[4]  Geoff Hulten,et al.  A General Framework for Mining Massive Data Streams , 2003 .

[5]  G. Lorden.  PROCEDURES FOR REACTING TO A CHANGE IN DISTRIBUTION , 1971 .

[6]  Anthony N. Pettitt,et al.  A simple cumulative sum type statistic for the change-point problem with zero-one observations , 1980 .

[7]  Douglas M. Hawkins,et al.  A Change-Point Model for a Shift in Variance , 2005 .

[8]  Eleftheriou Mavroudis,et al.  EWMA Control Charts for Monitoring High Yield Processes , 2013 .

[9]  Zachary G. Stoumbos,et al.  A CUSUM Chart for Monitoring a Proportion When Inspecting Continuously , 1999 .

[10]  A. Agresti.  [A Survey of Exact Inference for Contingency Tables]: Rejoinder , 1992 .

[11]  Fah Fatt Gan,et al.  CUMULATIVE SUM CHARTS FOR HIGH YIELD PROCESSES , 2001 .

[12]  W. John Braun.  Run length distributions for estimated attributes charts , 1999 .

[13]  David V. Hinkley,et al.  Corrections and AmendmentsBiometrika (1970), 57, 477-88: ‘Inference about the change-point in a sequence of binomial variables’ , 1971 .

[14]  Dimitris K. Tasoulis,et al.  Nonparametric Monitoring of Data Streams for Changes in Location and Scale , 2011, Technometrics.

[15]  Arjun K. Gupta,et al.  Testing and Locating Variance Changepoints with Application to Stock Prices , 1997 .

[16]  William H. Woodall,et al.  Control Charts Based on Attribute Data: Bibliography and Review , 1997 .

[17]  Arthur B. Yeh,et al.  EWMA control charts for monitoring high-yield processes based on non-transformed observations , 2008 .

[18]  Douglas M. Hawkins,et al.  The Changepoint Model for Statistical Process Control , 2003 .

[19]  Lloyd S. Nelson,et al.  A Control Chart for Parts-Per-Million Nonconforming Items , 1994 .

[20]  David V. Hinkley,et al.  Inference about the change-point in a sequence of binomial variables , 1970 .