Computing and approximating multivariate chi-square probabilities

Abstract

We consider computational methods for evaluating and approximating multivariate chi-square probabilities in cases where the pertaining correlation matrix or blocks thereof have a low factorial representation. To this end, techniques from matrix factorization and probability theory are applied. We outline a variety of statistical applications of multivariate chi-square distributions and provide a system of MATLAB programs implementing the proposed algorithms. Computer simulations demonstrate the accuracy and the computational efficiency of our methods in comparison with Monte Carlo approximations, and a real data example from statistical genetics illustrates their usage in practice.

Keywords:

• Bonferroni inequalities
• chain factorization
• correlation matrix
• effective number of tests
• linkage disequilibrium
• m-factorial matrix
• product-type probability approximations
• sub-Markovian
• Wishart matrix

AMS Subject Classification:

• 60E05
• 60E15
• 62E17
• 65D20

Acknowledgements

We are grateful to an anonymous referee for constructive comments which improved the presentation. We thank Thomas Royen for many valuable comments, especially regarding Remark 3.7.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partly supported by the Deutsche Forschungsgemeinschaft via grant No. DI 1723/3-1.