Comparison Analysis of the Wald-s and the Bayes Type Sequential Methods for Testing Hypotheses

Authors: K. J. Kachiashvili

Abstract:

The Comparison analysis of the Wald-s and Bayestype sequential methods for testing hypotheses is offered. The merits of the new sequential test are: universality which consists in optimality (with given criteria) and uniformity of decision-making regions for any number of hypotheses; simplicity, convenience and uniformity of the algorithms of their realization; reliability of the obtained results and an opportunity of providing the errors probabilities of desirable values. There are given the Computation results of concrete examples which confirm the above-stated characteristics of the new method and characterize the considered methods in regard to each other.

Keywords: Errors of types I and II, likelihood ratio, the Bayes Type Sequential test, the Wald's sequential test, averaged number of observations.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1328864

References:

[1] A. Wald, Sequential analysis. NY: Wiley, 1947.
[2] A. Wald, "Foundations of a General Theory of Sequential Decision Functions," Econometrica, vol. 15, pp. 279-313, 1947.
[3] G.A. Barnard, "Sequential test in industrial statistics," Suppl. J. Roy. Statist. Soc., vol. 8, no. 1, 1946.
[4] B.K. Ghosh, and P.K. Sen, (eds.) Handbook of Sequential Analysis. NY: Dekker, 1991.
[5] A. Wald, and J. Wolfowitz, "Optimum Character of the Sequential Probability Ratio Test," Ann. Math. Statist., vol. 19, pp. 326-339, 1948.
[6] M.A. Girshick, "Contributions of the Theory of Sequential Analysis," Ann. Math. Statist., vol 17, no. 2, pp. 123-143, 1946.
[7] M.A. Girshick, "Contributions of the Theory of Sequential Analysis," Ann. Math. Statist., vol. 17, no. 3, pp. 282-298, 1946.
[8] B.K. Ghosh, Sequential Tests of Statistical Hypotheses. Reading, Massachusetts: Addison-Wesley, 1970.
[9] R. Lepage, Smoothing Wald-s test. Technical Report, Department of Statistics, East Lansing: Michigan State University, 1979.
[10] J.O. Berger, and R. Wolpert, The Likelihood Principle. Institute of Mathematical Statistics Monograph Series, CA: Hayward, 1984.
[11] D. Siegmund, Sequential Analysis. NY: Springer Series in Statistics. Springer-Verlag, 1985.
[12] S.A. Aivazjan, "A comparison of the optimal properties of the Neuman- Pearson and the Wald sequential probability ratio test," Teoria Verojatn. i Primenen., vol. 4, no. 1, pp. 86-93, 1959.
[13] A. Gut, and R. Schwabe, "Some properties of multiple decision following a sequential test," Seq. Anal., vol. 15, pp. 199-310, 1996.
[14] A. Gut, and R. Schwabe, "Some remarks on repeated significance tests for linear contrasts," J. Stat. Plan. Inference, vol. 82, pp. 25-34, 1999.
[15] A. Irle, and V. Lotov, "On the properties of a multiple sequential test," Metrika, vol. 72, pp. 189-198, 2010.
[16] K.J. Arrow, D. Blackwell, and M.A. Girshick, "Bayes and minimax solutions of sequential decision problems," Econometrica, vol. 17, no. 2, pp. 213-244, 1949.
[17] G.A. Barnard, “A review of “Sequential Analysis” by Abraham Wald,” J. Amer. Statist. Assoc., vol. 42, pp. 658-669, 1947.
[18] G.A. Barnard, “Statistical inference (with discussion),” J. Roy. Statist. Soc. (Ser. B), vol. 11, pp. 115-139, 1949.
[19] S.J. Amster, “A modified Bayes stopping rule,” Ann. Math. Statist., vol. 34, no. 4, pp. 1404-1413, 1963.
[20] F.J. Anscombe, “Sequential medical trials,” J. Amer. Statist. Assoc., vol. 58, pp. 365-383, 1963.
[21] W. Edwards, H. Lindman, and L.J. Savage, “Bayesian statistics inference for psychological research,” Psychol. Rev., vol. 70, no. 3, pp. 193-242, 1963.
[22] S.N. Ray, “Bounds on the maximum sample size of a Bayes sequential procedure,” Ann. Math. Statist., vol. 36, no. 3, pp. 859-878, 1965.
[23] D.J. Bartholomew, “Hypothesis testing when the sample size is treated as a random variable,” J. Roy. Statist. Soc., vol. 29, pp. 53-82, 1967.
[24] D. Basu, “Statistical information and likelihood (with discussion),” Sankhya (Ser. A), vol. 37, no. 1, pp. 1-71, 1975.
[25] D.V. Lindley, and L.D. Phillips, Inference for a Bernoulli process (a Bayesian view),” Amer. Statist., vol. 30, pp. 112-119, 1976.
[26] J.O. Berger, Statistical Decision Theory and Bayesian Analysis. NY: Springer, 1985.
[27] K.J. Kachiashvili, and M.A. Hashmi, “About Using Sequential Analysis Approach for Testing Many Hypotheses,” Bulletin of the Georgian Academy of Sciences, vol. 4, no. 2, pp. 20-25, 2010.
[28] K.J. Kachiashvili, M. A. Hashmi, and A. Mueed, “Sensitivity Analysis of Classical and Conditional Bayesian Problems of Many Hypotheses Testing,” Communications in Statistics—Theory and Methods, vol. 41, no. 4, pp. 591–605, 2012.
[29] K.J. Kachiashvili, and A. Mueed, “Conditional Bayesian Task of Testing Many Hypotheses,” Statistics, 2011. DOI:10.1080/02331888.2011.602681.
[30] G.K. Kachiashvili, K.J. Kachiashvili, and A. Mueed, “Specific Features of Regions of Acceptance of Hypotheses in Conditional Bayesian Problems of Statistical Hypotheses Testing,” Sankhya, Series A, vol. 74 A1, 2012.
[31] M. Kendall, and A. Stuart, The Advanced Theory of Statistics. vol. 2., London: Charles Griffin & Company Limited, 1970.
[32] A. Stuart, J.K. Ord, and S. Arnold, Kendall’s Advanced Theory of Statistics. Vol. 2A: Classification Inference and the Linear Model. NY, 1999.
[33] K.J. Kachiashvili “Investigation and Computation of Unconditional and Conditional Bayesian Problems of Hypothesis Testing,” ARPN Journal of Systems and Software, vol.1, no.2, pp. 47-59, May 2011.
[34] S. Kullback, Informatin Theory and Statistics. Wiley, 1978.