Lobachevskii Journal of Mathematics

, Volume 39, Issue 3, pp 368–376 | Cite as

Locally Most Powerful Group-Sequential Tests with Groups of Observations of Random Size: Finite Horizon

  • A. Novikov
  • P. Novikov


We consider sequential hypothesis testing based on observations which are received in groups of random size. The observations are supposed independent both within and between the groups, with a distribution depending on a real-valued parameter θ. We suppose that the group sizes are independent and their distributions are known, and that the groups are formed independently from the observations. We are concerned with a problem of testing a simple hypothesis H0: θ = θ0 against a composite alternative H1: θ > θ0, supposing that no more than a given number of groups will be available (finite horizon). For any (group-)sequential test, we take into account the following three characteristics: its error probability of the first type, the derivative of its power function at θ = θ0, and the average cost of observations, under some natural assumptions about the cost structure. Under suitable regularity conditions, we characterize the structure of all sequential tests maximizing the derivative of the power function among all (finite-horizon) sequential tests whose error probability of the first type and the average cost of observations do not exceed some prescribed levels.


Sequential analysis hypothesis testing locally most powerful test optimal stopping optimal sequential tests 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. H. Berk, “Locally most powerful sequential tests,” Ann. Stat. 3, 373–381 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    N. Mukhopadhyay and B. M. de Silva, “Theory and applications of a new methodology for the random sequential probability ratio test,” Stat. Methodol. 5, 424–453 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Novikov and P. Novikov, “Locally most powerful sequential tests of a simple hypothesis vs. one-sided alternatives,” J. Stat. Planning Inference 140, 750–765 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    An. A. Novikov and P. A. Novikov, “Locally most powerful sequential tests of a simple hypothesis vs. one-sided alternatives for independent observations,” Theory Probab. Appl. 56, 449–477 (2011).zbMATHGoogle Scholar
  5. 5.
    An. A. Novikov and P. A. Novikov, “Information inequalities for characteristics of group-sequential test with groups of observations of random size,” Russ. Math. (Iz. VUZ) 60 (12), 54–61 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. T. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, New Jersey, 1970).CrossRefzbMATHGoogle Scholar
  7. 7.
    M. Roters, “Locally most powerful sequential tests for processes of the exponential class with stationary and independent increments,” Metrika 39, 177–183 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    N. Schmitz, Optimal Sequentially Planned Decision Procedures (Springer, New York, 1993).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsMetropolitan Autonomous University—IztapalapaMexico CityMexico
  2. 2.Kazan Federal UniversityHigher Institute for Information Technology and Information SystemsKazanRussia

Personalised recommendations