A Decomposition of Pearson–Fisher and Dzhaparidze–Nikulin Statistics and Some Ideas for a More Powerful Test Construction

Abstract

An explicit decomposition on asymptotically independent distributed as chi-squared with one degree of freedom components of the Pearson–Fisher and Dzhaparidze–Nikulin tests is presented. The decomposition is formally the same for both tests and is valid for any partitioning of a sample space. Vector-valued tests, components of which can be not only different scalar tests based on the same sample, but also scalar tests based on components or groups of components of the same statistic are considered. Numerical examples illustrating the idea are presented.

Keywords:

• Decomposition of tests
• Dzhaparidze–Nikulin test
• Pearson–Fisher test
• Power of components
• Vector-valued tests

Mathematics Subject Classification:

• 62F03
• 62F05

Acknowledgment

The author would like to thank N. Balakrishnan and the referees for their help in improving the presentation.