The Annals of Statistics

Asymptotically optimal multistage tests of simple hypotheses

Jay Bartroff

Full-text: Open access


A family of variable stage size multistage tests of simple hypotheses is described, based on efficient multistage sampling procedures. Using a loss function that is a linear combination of sampling costs and error probabilities, these tests are shown to minimize the integrated risk to second order as the costs per stage and per observation approach zero. A numerical study shows significant improvement over group sequential tests in a binomial testing problem.

Article information

Ann. Statist., Volume 35, Number 5 (2007), 2075-2105.

First available in Project Euclid: 7 November 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L10: Sequential analysis
Secondary: 91A20: Multistage and repeated games

Multistage hypothesis testing asymptotic optimality group sequential


Bartroff, Jay. Asymptotically optimal multistage tests of simple hypotheses. Ann. Statist. 35 (2007), no. 5, 2075--2105. doi:10.1214/009053607000000235.

Export citation


  • Barber, S. and Jennison, C. (2002). Optimal asymmetric one-sided group sequential tests. Biometrika 89 49--60.
  • Bartroff, J. (2004). Asymptotically optimal multistage hypothesis tests. Ph.D. dissertation, Caltech.
  • Bartroff, J. (2006). Optimal multistage sampling in a boundary-crossing problem. Sequential Anal. 25 59--84.
  • Chernoff, H. (1961). Sequential tests for the mean of a normal distribution. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 79--91. Univ. California Press, Berkeley.
  • Cressie, N. and Morgan, P. B. (1993). The VPRT: A sequential testing procedure dominating the SPRT. Econometric Theory 9 431--450.
  • Durrett, R. (1995). Probability: Theory and Examples, 2nd ed. Duxburry, Belmont, CA.
  • Eales, J. D. and Jennison, C. (1992). An improved method for deriving optimal one-sided group sequential tests. Biometrika 79 13--24.
  • Eales, J. D. and Jennison, C. (1995). Optimal two-sided group sequential tests. Sequential Anal. 14 273--286.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • Hall, P. (1982). Rates of Convergence in the Central Limit Theorem. Pitman, Boston.
  • Jennison, C. and Turnbull, B. W. (2000). Group Sequential Methods with Applications to Clinical Trials. Chapman and Hall/CRC, Boca Raton, FL.
  • Kim, K. and DeMets, D. L. (1987). Design and analysis of group sequential tests based on the type I error spending rate function. Biometrika 74 149--154.
  • Lai, T. L. and Shih, M.-C. (2004). Power, sample size and adaptation considerations in the design of group sequential clinical trials. Biometrika 91 507--528.
  • Lorden, G. (1967). Integrated risk of asymptotically Bayes sequential tests. Ann. Math. Statist. 38 1399--1422.
  • Lorden, G. (1976). 2-SPRT's and the modified Kiefer--Weiss problem of minimizing an expected sample size. Ann. Statist. 4 281--291.
  • Lorden, G. (1977). Nearly-optimal sequential tests for finitely many parameter values. Ann. Statist. 5 1--21.
  • Lorden, G. (1983). Asymptotic efficiency of three-stage hypothesis tests. Ann. Statist. 11 129--140.
  • Morgan, P. B. and Cressie, N. (1997). A comparison of the cost-efficiencies of the sequential, group-sequential and variable-sample-size-sequential probability ratio tests. Scand. J. Statist. 24 181--200.
  • Pocock, S. J. (1982). Interim analyses for randomized clinical trials: The group sequential approach. Biometrics 38 153--162.
  • Schmitz, N. (1993). Optimal Sequentially Planned Decision Procedures. Lecture Notes in Statist. 79. Springer, New York.
  • Wang, S. K. and Tsiatis, A. A. (1987). Approximately optimal one-parameter boundaries for group sequential trials. Biometrics 43 193--199.