Second-Order Optimal Sequential Tests

  • M. B. Malyutov
  • I. I. Tsitovich
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 51)


An asymptotic lower bound is derived involving a second additive term of order \(\sqrt {\left| {\ln \alpha } \right|} \) as α → 0 for the mean length of a controlled sequential strategy s for discrimination between two statistical models in a very general nonparametric setting. The parameter a is the maximal error probability of s.

A sequential strategy is constructed attaining (or almost attaining) this asymptotic bound uniformly over the distributions of models including those from the indifference zone. These results are extended for a general loss function g(N) with the power growth of the strategy length N.

Applications of these results to change-point detection and testing homogeneity are outlined.


change-point detection controlled experiments general risk second-order optimality sequential test 


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  1. Barron, A.R. and Sheu, C.-H. (1991). Approximation of density functions by sequences of exponential families. Ann. Statist. 19, 1347–1369.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Centsov, N.N. (1982). Statistical Decision Rules and Optimal Inference. Amer. Math. Soc. Transl. 53. Providence RI.Google Scholar
  3. Chernoff, H. (1959). Sequential design of experiments. Ann. Math. Statist. 30, 755–770.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Chernoff, H. (1997). Sequential Analysis and Optimal Design. Philadelphia: SIAM.Google Scholar
  5. Keener, R. (1984). Second order efficiency in the sequential design of experiments. Ann. Statist. 12, 510–532.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Lai, T.L. (1995). Sequential change-point detection in quality control and dynamical systems. J. Roy. Statist. Soc. B 57, 613–658.zbMATHGoogle Scholar
  7. Lalley, S.P. and Lorden, G. (1986). A control problem arising in the sequential design of experiments. Ann. Probab. 14, 136–172.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Lorden, G. (1971). Procedures for reacting to a change in distribution. Ann. Math. Statist. 42, 1897–1908.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Malyutov, M.B. (1983). Lower bounds for the mean length of sequentially designed experiments. Soviet Math. (Izv. VUZ.) 27, 21–47.zbMATHGoogle Scholar
  10. Malyutov, M.B. and Tsitovich, I.I. (1997a). Asymptotically optimal sequential testing of hypotheses. In Proc. Internat. Conf. on Distributed Computer Communication Networks: Theory and Applications, pp. 134–141. Tel-Aviv.Google Scholar
  11. Malyutov, M.B. and Tsitovich, I.I. (1997b). Sequential search for significant variables of an unknown function. Problems of Inform. Transmiss. 33, 88–107.MathSciNetGoogle Scholar
  12. Malyutov, M.B. and Tsitovich, I.I. (2000). Asymptotically optimal sequential testing of hypotheses. Problems of Inform. Transmiss. (To appear).Google Scholar
  13. Schwarz, G. (1962). Asymptotic shapes of Bayes sequential testing regions. Ann. Math. Statist. 33, 224–236.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Tsitovich, I.I. (1984). On sequential design of experiments for hypothesis testing. Theory Probab. and Appl. 29, 778–781.MathSciNetzbMATHGoogle Scholar
  15. Tsitovich, I.I. (1990). Sequential design and discrimination. In Models and Methods of Information Systems, pp. 36–48. Moscow: Nauka (in Russian).Google Scholar
  16. Tsitovich, I.I. (1993). Sequential Discrimination. D.Sci. thesis. Moscow (in Russian).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • M. B. Malyutov
    • 1
  • I. I. Tsitovich
    • 2
  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Institute for Information Transmission ProblemsMoscowRussia

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