Specific features of regions of acceptance of hypotheses in conditional Bayesian problems of statistical hypotheses testing Authors G. K. Kachiashvili Georgian Technical University K. J. Kachiashvili I. Vekua Institute of Applied Mathematics of Tbilisi State University Abdus Salam School of Mathematical Sciences of GC University A. Mueed Air University Multan Campus Article

First Online: 13 November 2012 Received: 26 October 2010 Revised: 09 August 2011 DOI :
10.1007/s13171-012-0014-8

Cite this article as: Kachiashvili, G.K., Kachiashvili, K.J. & Mueed, A. Sankhya A (2012) 74: 112. doi:10.1007/s13171-012-0014-8
Abstract Specific features of the regions of acceptance of hypotheses in conditional Bayesian problems of statistical hypotheses testing are discussed. It is shown that the classical Bayesian statement of the problem of statistical hypotheses testing in the form of an unconditional optimizing problem is a special case of conditional Bayesian problems of hypotheses testing set in the form of conditional optimizing problems. It is also shown that, at acceptance of hypotheses in conditional problems of hypotheses testing, the situation is similar to the sequential analysis. It is possible an occurrence of the situation when the acceptance of a hypothesis with specified validity on the basis of the available information is impossible. In such a situation, the actions are similar to the sequential analysis, i.e. it is necessary to obtain additional information in the form of new observation results or to change the significance level of a test.

Keywords and phrases Bayesian problem hypotheses testing significance level conditional problem unconditional problem

References Berger, J.O. (1985).

Statistical decision theory and bayesian analysis . Springer, New York.

MATH Bernardo, J.M. and Smith, A.F.M. (1996). Bayesian theory . Wiley, New York

Bhattacharyya, A. (1943). On a measure of divergence between two statistical populations defined by their probability distributions.

Bull. Calcutta Math. Soc.,
35 , 99–110.

MathSciNet MATH Casella, G. and Berger, R.L. (1987). Reconciling Bayesian and frequentist evidence in the one-sided testing problem (with comments).

J. Amer. Statist. Assoc.,
82 , 106–111.

MathSciNet MATH CrossRef De Groot, M. (1970). Optimal statistical decisions . McGraw-Hill Book Company, New York.

Duda, R.O., Hart, P.E. and Stork, D.G. (2006). Pattern classification , (Second Edition). Wiley, Singapore.

Kachiashvili, K.J. (1989). Bayesian algorithms of many hypothesis testing . Ganatleba, Tbilisi.

Kachiashvili, K.J. (2003). Generalization of Bayesian rule of many simple hypotheses testing.

International Journal of Information Technology &

Decision Making, World Scientific Publishing Company,
2 , 41–70.

CrossRef Kachiashvili, K.J., Hashmi, M.A. and Mueed, A. (2012). Sensitivity analysis of classical and conditional Bayesian problems of many hypotheses testing.

Comm. Statist. Theory Methods,
41 , 591–605. DOI: 10.1080/03610926.2010.510255.

MathSciNet MATH CrossRef Lee, P.M. (1989).

Bayesian statistics: an introduction . Edward Arnold, London.

MATH Lehmann, E.L. (1986). Testing Statistical hypotheses, (2nd ed.). Wiley, New York.

Lindley, D.V. (1991). Making decision . Wiley, New York.

Meng, C.Y.K. and Dempster, A.P. (1987). A Bayesian approach to the multiplicity problem for significance testing with binomial data.

Biometrics,
43 , 301–311.

MathSciNet CrossRef Moreno, E. and Cano, J.A. (1989). Testing a point null hypothesis: asymptotic robust Bayesian analysis with respect to the priors given on a subsigma field.

Int. Statist. Rev.,
57 , 221–232.

MATH CrossRef Sage, A.P. and Melse, J.L. (1972). Estimation theory with application to communication and control . McGraw-Hill, New York.

Stuart, A., Ord, J.K. and Arnold, S. (1999), Kendall’s advanced theory of statistics . Classical Inference and The Linear Model, vol. 2A, (Sixth Edition). Oxford University Press Inc., New York.

Wald, A. (1947).

Sequential analysis . Wiley, New York.

MATH Wald, A. (1950a).

Statistical decision functions . Wiley, New York.

MATH Wald, A. (1950b). Basic ideas of a general theory of statistical decision rules, vol. I. In Proceedings of the International Congress of Mathematicians .

Westfall, P.H., Johnson, W.O. and Utts, J.M. (1997). A Bayesian perspective on the Bonferroni adjustment.

Biometrika ,

84 , 419–427.

MathSciNet MATH CrossRef Zacks, S. (1971). The theory of statistical inference . Wiley, New York.

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