Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 356587, 8 pages
http://dx.doi.org/10.1155/2015/356587
Research Article

Asymptotic Optimality of Combined Double Sequential Weighted Probability Ratio Test for Three Composite Hypotheses

School of Finance and Statistics, East China Normal University, Shanghai 200241, China

Received 24 December 2014; Revised 13 March 2015; Accepted 15 March 2015

Academic Editor: Antonino Laudani

Copyright © 2015 Lei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. J. Goeman, A. Solari, and T. Stijnen, “Three-sided hypothesis testing: simultaneous testing of superiority, equivalence and inferiority,” Statistics in Medicine, vol. 29, no. 20, pp. 2117–2125, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. K. S. Fu, Sequential Methods in Pattern Recognition and Learning, Academic Press, New York, NY, USA, 1968.
  3. T. McMillen and P. Holmes, “The dynamics of choice among multiple alternatives,” Journal of Mathematical Psychology, vol. 50, no. 1, pp. 30–57, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. J. J. Bussgang, “Sequential methods in radar detection,” Proceedings of the IEEE, vol. 58, no. 5, pp. 731–743, 1970. View at Publisher · View at Google Scholar
  5. S. L. Anderson, “Simple method of comparing the breaking strength of two yarns,” Journal of the Textle Institute, vol. 45, pp. 472–479, 1954. View at Google Scholar
  6. Y. Li, X. L. Pu, and F. Tsung, “Adaptive charting schemes based on double sequential probability ratio tests,” Quality and Reliability Engineering International, vol. 25, no. 1, pp. 21–39, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. I. V. Pavlov, “A sequential procedure for testing many composite hypotheses,” Theory of Probability & Its Applications, vol. 32, no. 1, pp. 138–142, 1988. View at Publisher · View at Google Scholar
  8. C. W. Baum and V. V. Veeravalli, “A sequential procedure for multihypothesis testing,” IEEE Transactions on Information Theory, vol. 40, no. 6, pp. 1994–2007, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. V. P. Dragalin, A. G. Tartakovsky, and V. V. Veeravalli, “Multihypothesis sequential probability ratio tests, I: asymptotic optimality,” IEEE Transactions on Information Theory, vol. 45, no. 7, pp. 2448–2461, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. V. P. Dragalin, A. G. Tartakovsky, and V. V. Veeravalli, “Multihypothesis sequential probability ratio tests: II. Accurate asymptotic expansions for the expected sample size,” IEEE Transactions on Information Theory, vol. 46, no. 4, pp. 1366–1383, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. M. Sobel and A. Wald, “A sequential decision procedure for choosing one of three hypotheses concerning the unknown mean of a normal distribution,” Annals of Mathematical Statistics, vol. 20, pp. 502–522, 1949. View at Publisher · View at Google Scholar · View at MathSciNet
  12. P. Armitage, “Sequential analysis with more than two alternative hypotheses, and its relation to discriminant function analysis,” Journal of the Royal Statistical Society. Series B. Methodological, vol. 12, pp. 137–144, 1950. View at Google Scholar · View at MathSciNet
  13. G. Simons, “Lower bounds for average sample number of sequential multihypothesis tests,” Annals of Mathematical Statistics, vol. 38, pp. 1343–1364, 1967. View at Publisher · View at Google Scholar · View at MathSciNet
  14. G. Lorden, “Likelihood ratio tests for sequential k-decision problems,” Annals of Mathematical Statistics, vol. 43, pp. 1412–1427, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  15. J. Whitehead and H. Brunier, “The double triangular test: a sequential test for the two-sided alternative with early stopping under the null hypothesis,” Sequential Analysis: Design Methods & Applications, vol. 9, no. 2, pp. 117–136, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Y. Li and X. Pu, “Hypothesis designs for three-hypothesis test problems,” Mathematical Problems in Engineering, vol. 2010, Article ID 393095, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. Y. Li and X. L. Pu, “A method for designing three-hypothesis test problems and sequential schemes,” Communications in Statistics—Simulation and Computation, vol. 39, no. 9, pp. 1690–1708, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. V. P. Dragalin and A. A. Novikov, “Adaptive sequential tests for composite hypotheses,” Surveys of Applied and Industrial Mathematics, vol. 6, pp. 387–398, 1999. View at Google Scholar
  19. T. L. Lai, “Sequential multiple hypothesis testing and efficient fault detection-isolation in stochastic systems,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 595–608, 2000. View at Publisher · View at Google Scholar · View at Scopus
  20. L. Wang, D. D. Xiang, X. L. Pu, and Y. Li, “A double sequential weighted probability ratio test for one-sided composite hypotheses,” Communications in Statistics. Theory and Methods, vol. 42, no. 20, pp. 3678–3695, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. G. Lorden, “2-SPRT's and the modified Kiefer-Weiss problem of minimizing an expected sample size,” The Annals of Statistics, vol. 4, no. 2, pp. 281–291, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. D. Chen and F. J. Hickernell, “A class of asymptotically optimal sequential tests for composite hypotheses,” Science in China Series A, vol. 37, no. 11, pp. 1314–1324, 1994. View at Google Scholar · View at MathSciNet
  23. W. Hoeffding, “Lower bounds for the expected sample size and the average risk of a sequential procedure,” Annals of Mathematical Statistics, vol. 31, pp. 352–368, 1960. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. D. Chen, “Asymptotic optimality for one class of truncated sequential tests,” Science in China. Series A. Mathematics, Physics, Astronomy, vol. 30, pp. 30–41, 1991. View at Google Scholar · View at MathSciNet