Annals of Statistics

Nonanticipating estimation applied to sequential analysis and changepoint detection

Gary Lorden and Moshe Pollak

Full-text: Open access


Suppose a process yields independent observations whose distributions belong to a family parameterized by θ∈Θ. When the process is in control, the observations are i.i.d. with a known parameter value θ0. When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37–41] to construct a class of sequential tests and detection schemes whereby the unknown post-change parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixture-type rules are operationally or conceptually difficult to formulate. We exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a Gamma distribution, in both a univariate and a multivariate setting.

Article information

Ann. Statist., Volume 33, Number 3 (2005), 1422-1454.

First available in Project Euclid: 1 July 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L10: Sequential analysis 62N10 62F03: Hypothesis testing
Secondary: 62F05: Asymptotic properties of tests 60K05: Renewal theory

Quality control cusum Shiryayev–Roberts surveillance statistical process control power one tests renewal theory nonlinear renewal theory Gamma distribution


Lorden, Gary; Pollak, Moshe. Nonanticipating estimation applied to sequential analysis and changepoint detection. Ann. Statist. 33 (2005), no. 3, 1422--1454. doi:10.1214/009053605000000183.

Export citation


  • Dragalin, V. P. (1997). The sequential change point problem. Economic Quality Control 12 95–122.
  • Lai, T. L. and Siegmund, D. (1977). A nonlinear renewal theory with applications to sequential analysis. I. Ann. Statist. 5 946–954.
  • Lorden, G. and Pollak, M. (1994). An alternative to mixtures for sequential testing and changepoint detection. Technical report.
  • Pollak, M. (1978). Optimality and almost optimality of mixture stopping rules. Ann. Statist. 6 910–916.
  • Pollak, M. (1987). Average run lengths of an optimal method of detecting a change in distribution. Ann. Statist. 15 749–779.
  • Pollak, M. and Siegmund, D. (1975). Approximations to the expected sample size of certain sequential tests. Ann. Statist. 3 1267–1282.
  • Pollak, M. and Siegmund, D. (1991). Sequential detection of a change in a normal mean when the initial value is unknown. Ann. Statist. 19 394–416.
  • Pollak, M. and Yakir, B. (1999). A simple comparison of mixture vs. nonanticipating estimation. Sequential Anal. 18 157–164.
  • Robbins, H. and Siegmund, D. (1972). A class of stopping rules for testing parametric hypotheses. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 37–41. Univ. California Press, Berkeley.
  • Robbins, H. and Siegmund, D. (1974). The expected sample size of some tests of power one. Ann. Statist. 2 415–436.
  • Roberts, S. W. (1966). A comparison of some control chart procedures. Technometrics 8 411–430.
  • Shiryayev, A. N. (1963). On optimum methods in quickest detection problems. Theory Probab. Appl. 8 22–46.
  • Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.
  • Siegmund, D. (1986). Boundary crossing probabilities and statistical applications. Ann. Statist. 14 361–404.
  • Siegmund, D. and Venkatraman, E. S. (1995). Using the generalized likelihood ratio statistic for sequential detection of a change-point. Ann. Statist. 23 255–271.
  • Stone, C. (1965). On moment generating functions and renewal theory. Ann. Math. Statist. 36 1298–1301.
  • Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia.
  • Yakir, B. (1995). A note on the run length to false alarm of a change-point detection policy. Ann. Statist. 23 272–281.
  • Yakir, B. and Pollak, M. (1998). A new representation for a renewal-theoretic constant appearing in asymptotic approximations of large deviations. Ann. Appl. Probab. 8 749–774.