Skip Main Navigation Skip To Navigation Content

Research Reports

You are here: Census.govSubjects A to ZResearch Reports Sorted by Year › Abstract of RRS2012/15
Skip top of page navigation

The Multiple Testing Problem for Box-Pierce Statistics

Tucker McElroy and Brian Monsell


ARIMA models, Ljung-Box statistic, Time series residuals.


We derive the exact joint asymptotic distribution for multiple Box-Pierce statistics, and use these results to determine appropriate critical values in joint testing problems of time series goodness-of-fit. A novel alpha-rationing scheme, motivated by the sequence of conditional probabilities for the statistical tests, is developed and implemented. This method can be used to produce critical values and p-values both for each step of the sequential testing procedure, and for the procedure as a whole. Efficient computational algorithms are discussed. Simulation studies assess the impact of finite samples on the real Type I error. It is also demonstrated empirically that the conventional x-squared critical values for the Box-Pierce statistics are too small, with a Type I error rate greater than the nominal; the new method does not suffer from this defect, and allows for more rigorous model-building.


Tucker McElroy and Brian Monsell. (2012). The Multiple Testing Problem for Box-Pierce Statistics. Center for Statistical Research & Methodology Research Report Series (Statistics #2012-15). U.S. Census Bureau. Available online at <>.

Source: U.S. Census Bureau, Center for Statistical Research & Methodology, Research and Methodology Directorate

Published online: September 24, 2012
Last revised: September 24, 2012


[PDF] or PDF denotes a file in Adobe’s Portable Document Format. To view the file, you will need the Adobe® Reader® Off Site available free from Adobe.

This symbol Off Site indicates a link to a non-government web site. Our linking to these sites does not constitute an endorsement of any products, services or the information found on them. Once you link to another site you are subject to the policies of the new site.

Source: U.S. Census Bureau | Research and Methodology Directorate | Center for Statistical Research & Methodology | (301) 763-3215 (or |   Last Revised: September 11, 2013