Constrained Bayesian Method of Composite Hypotheses Testing: Singularities and Capabilities

Authors

  • K.J. Kachiashvili Georgian Technical University, 77, st. Kostava, Tbilisi, 0175, I. Vekua Institute of Applied Mathematics of the Tbilisi State University, 2, st. University, Tbilisi, 0179, Georgia

DOI:

https://doi.org/10.6000/1929-6029.2016.05.03.1

Keywords:

CBM, Bayesian test, Composite hypotheses, Hypotheses testing, Lindley's paradox

Abstract

The paper deals with the constrained Bayesian Method (CBM) for testing composite hypotheses. It is shown that, similarly to the cases when CBM is optimal for testing simple and multiple hypotheses in parallel and sequential experiments, it keeps the optimal properties at testing composite hypotheses. In particular, it easily, without special efforts, overcomes the Lindley’s paradox arising when testing a simple hypothesis versus a composite one. The CBM is compared with Bayesian test in the classical case and when the a priori probabilities are chosen in a special manner for overcoming the Lindley’s paradox. Superiority of CBM against these tests is demonstrated by simulation. The justice of the theoretical judgment is supported by many computation results of different characteristics of the considered methods.

References

Kachiashvili KJ, Hashmi MA, Mueed A. Sensitivity Analysis of Classical and Conditional Bayesian Problems of Many Hypotheses Testing. Communications in Statistics—Theory and Methods 2012; 41: 591-605. http://dx.doi.org/10.1080/03610926.2010.510255 DOI: https://doi.org/10.1080/03610926.2010.510255

Kachiashvili KJ. Comparison of Some Methods of Testing Statistical Hypotheses. (Part I. Parallel Methods and Part II. Sequential Methods). International Journal of Statistics in Medical Research 2014; 3: 174-197. http://dx.doi.org/10.6000/1929-6029.2014.03.02.11 DOI: https://doi.org/10.6000/1929-6029.2014.03.02.11

Kachiashvili KJ. Generalization of Bayesian Rule of Many Simple Hypotheses Testing. International Journal of Information Technology and Decision Making 2003; 2: 41-70. http://dx.doi.org/10.1142/S0219622003000525 DOI: https://doi.org/10.1142/S0219622003000525

Kachiashvili KJ. Investigation and Computation of Unconditional and Conditional Bayesian Problems of Hypothesis Testing. ARPN Journal of Systems and Software 2011; 1: 47-59.

Kachiashvili KJ. The Methods of Sequential Analysis of Bayesian Type for the Multiple Testing Problem. Sequential Analysis 2014; 33: 23-38. http://dx.doi.org/10.1080/07474946.2013.843318 DOI: https://doi.org/10.1080/07474946.2013.843318

Kachiashvili KJ, Hashmi MA. About Using Sequential Analysis Approach for Testing Many Hypotheses. Bulletin of the Georgian Academy of Sciences 2010; 4: 20-25.

Kachiashvili KJ, Hashmi MA, Mueed A. Quasi-optimal Bayesian procedures of many hypotheses testing. Journal of Applied Statistics 2013; 40: 103-122. http://dx.doi.org/10.1080/02664763.2012.734797 DOI: https://doi.org/10.1080/02664763.2012.734797

Kachiashvili GK, Kachiashvili KJ, Mueed A. Specific Features of Regions of Acceptance of Hypotheses in Conditional Bayesian Problems of Statistical Hypotheses Testing. Sankhya 2012; 74: 112-125. http://dx.doi.org/10.1007/s13171-012-0014-8 DOI: https://doi.org/10.1007/s13171-012-0014-8

Kachiashvili KJ, Mueed A. Conditional Bayesian Task of Testing Many Hypotheses. Statistics 2013; 47: 274-293. http://dx.doi.org/10.1080/02331888.2011.602681 DOI: https://doi.org/10.1080/02331888.2011.602681

Marden JI. Hypothesis Testing: From p Values to Bayes Factors, Am Stat Assoc 2000; 95: 1316-1320. http://dx.doi.org/10.2307/2669779 DOI: https://doi.org/10.1080/01621459.2000.10474339

Andersson S. Distributions of Maximal Invariants Using Quotient Measures. Ann Statist 1982; 10: 955-961. http://dx.doi.org/10.1214/aos/1176345885 DOI: https://doi.org/10.1214/aos/1176345885

Wijsman RA. Cross-Sections of Orbits and Their Application to Densities of Maximal Invariants, Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, 1; 1967; pp. 389-400.

Jeffreys H. Theory of Probability, Oxford: Clarendon Press 1939.

Berger JO, Pericchi LR. The Intrinsic Bayes Factor for Model Selection and Prediction, Am Stat Assoc 1996; 91: 109-122. http://dx.doi.org/10.1080/01621459.1996.10476668 DOI: https://doi.org/10.1080/01621459.1996.10476668

Kass RE, Wasserman L. The Selection of Prior Distributions by Formal Rules, Am Stat Assoc 1996; 91: 1343-1370. http://dx.doi.org/10.1080/01621459.1996.10477003 DOI: https://doi.org/10.1080/01621459.1996.10477003

Berger JO, Brown LD, Wolpert RL. A unified conditional frequentist and Bayesian test for fixed and sequential simple hypothesis testing. Ann Statist 1994; 22: 1787-1907. http://dx.doi.org/10.1214/aos/1176325757 DOI: https://doi.org/10.1214/aos/1176325757

Berger JO, Boukai B, Wang Y. Unified Frequentist and Bayesian Testing of a Precise Hypothesis. Statistical Science 1997; 12: 133-148. DOI: https://doi.org/10.1214/ss/1030037904

Berger JO, Boukai B, Wang Y. Simultaneous Bayesian-frequentist sequential testing of nested hypothes. Biometrika 1999; 86: 79-92. http://dx.doi.org/10.1093/biomet/86.1.79 DOI: https://doi.org/10.1093/biomet/86.1.79

Dass SC, Berger JO. Unified Conditional Frequentist and Bayesian Testing of Composite Hypotheses. Scand J of Stat 2003, 30: 193-210. http://dx.doi.org/10.1111/1467-9469.00326 DOI: https://doi.org/10.1111/1467-9469.00326

Gomez-Villegas MA, Main P, Sanz L. A Bayesian Analysis For The Multivariate Point Null Testing Problem. Statistics 2009; 43: 379-391. http://dx.doi.org/10.1080/02331880802505173 DOI: https://doi.org/10.1080/02331880802505173

Duchesne P, Francq Ch. Multivariate hypothesis testing using generalized and {2}-inverses – with applications. Statistics 2014; 1-22. DOI: https://doi.org/10.1080/02331888.2014.896917

Bedbur S, Beutner E, Kamps U. Multivariate testing and model-checking for generalized order statistics with applications. Statistics 2013: 1-114. DOI: https://doi.org/10.1080/02331888.2013.841696

Bernardo JM. A Bayesian analysis of classical hypothesis testing, Universidad de Valencia 1980; 605-617. DOI: https://doi.org/10.1007/BF02888370

Bartlett MS. A comment on D.V. Lindley’s statistical paradox. Biometrica 1957; 44: 533-534. http://dx.doi.org/10.1093/biomet/44.3-4.533 DOI: https://doi.org/10.1093/biomet/44.3-4.533

Demster AP. Model searching and estimation in the logic of inference. In Godambe VP, Sprott DA, editors. Foundations of Statistical Inference. Toronto: Holt, Rinehart and Winston 1971; pp. 56-81.

Moreno E, Giron FJ. On the frequentist and Bayesian approaches to hypothesis testing. University of Granada and University of Malaga 2006; SORT 30: 3-28.

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Published

2016-08-16

How to Cite

Kachiashvili, K. (2016). Constrained Bayesian Method of Composite Hypotheses Testing: Singularities and Capabilities. International Journal of Statistics in Medical Research, 5(3), 135–167. https://doi.org/10.6000/1929-6029.2016.05.03.1

Issue

Section

Special Issue - Inference in Clinical Experiments