The Annals of Statistics

Optimality of the CUSUM procedure in continuous time

George V. Moustakides

Full-text: Open access


The optimality of CUSUM under a Lorden-type criterion setting is considered. We demonstrate the optimality of the CUSUM test for Itô processes, in a sense similar to Lorden's, but with a criterion that replaces expected delays by the corresponding Kullback-Leibler divergence.

Article information

Ann. Statist., Volume 32, Number 1 (2004), 302-315.

First available in Project Euclid: 12 March 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L10: Sequential analysis
Secondary: 62L15: Optimal stopping [See also 60G40, 91A60] 62C10: Bayesian problems; characterization of Bayes procedures

CUSUM change point disorder problem sequential detection Kullback-Leibler divergence


Moustakides, George V. Optimality of the CUSUM procedure in continuous time. Ann. Statist. 32 (2004), no. 1, 302--315. doi:10.1214/aos/1079120138.

Export citation


  • Beibel, M. (1996). A note on Ritov's Bayes approach to the minmax property of the CUSUM procedure. Ann. Statist. 24 1804--1812.
  • Irle, A. (1984). Extended optimality of sequential probability ratio tests. Ann. Statist. 12 380--386.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Liptser, R. S. and Shiryayev, A. N. (1978). Statistics of Random Processes. 2. Applications. Springer, New York.
  • Lorden, G. (1971). Procedures for reacting to a change in distribution. Ann. Math. Statist. 42 1897--1908.
  • Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist. 14 1379--1387.
  • Moustakides, G. V. (1998). Quickest detection of abrupt changes for a class of random processes. IEEE Trans. Inform. Theory 44 1965--1968.
  • Øksendal, B. K. (1998). Stochastic Differential Equations: An Introduction with Applications, 5th ed. Springer, Berlin.
  • Page, E. S. (1954). Continuous inspection schemes. Biometrika 41 100--115.
  • Poor, H. V. (1998). Quickest detection with exponential penalty for delay. Ann. Statist. 26 2179--2205.
  • Ritov, Y. (1990). Decision theoretic optimality of the CUSUM procedure. Ann. Statist. 18 1464--1469.
  • Shiryayev, A. N. (1978). Optimal Stopping Rules. Springer, New York.
  • Shiryayev, A. N. (1996). Minimax optimality of the method of cumulative sums (CUSUM) in the case of continuous time. Russian Math. Surveys 51 750--751.
  • Tartakovski, A. (1995). Asymptotic properties of CUSUM and Shiryayev's procedures for detecting a change in a nonhomogeneous Gaussian process. Math. Methods Statist. 4 389--404.
  • Wald, A. (1947). Sequential Analysis. Wiley, New York.
  • Wald, A. and Wolfowitz, J. (1948). Optimum character of the sequential probability ratio test. Ann. Math. Statist. 19 326--339.
  • Yashin, A. (1983). On a problem of sequential hypothesis testing. Theory Probab. Appl. 28 157--165.