The Annals of Statistics

Quickest detection with exponential penalty for delay

H. Vincent Poor

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The problem of detecting a change in the probability distribution of a random sequence is considered. Stopping times are derived that optimize the tradeoff between detection delay and false alarms within two criteria. In both cases, the detection delay is penalized exponentially rather than linearly, as has been the case in previous formulations of this problem. The first of these two criteria is to minimize a worst-case measure of the exponential detection delay within a lower-bound constraint on the mean time between false alarms. Expressions for the performance of the optimal detection rule are also developed for this case. It is seen, for example, that the classical Page CUSUM test can be arbitrarily unfavorable relative to the optimal test under exponential delay penalty. The second criterion considered is a Bayesian one, in which the unknown change point is assumed to obey a geometric prior distribution. In this case, the optimal stopping time effects an optimal trade-off between the expected exponential detection delay and the probability of false alarm. Finally, generalizations of these results to problems in which the penalties for delay may be path dependent are also considered.

Article information

Ann. Statist., Volume 26, Number 6 (1998), 2179-2205.

First available in Project Euclid: 21 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L10: Sequential analysis
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62L15 94A13: Detection theory

Quickest detection change point problems optimal stopping exponential cost


Poor, H. Vincent. Quickest detection with exponential penalty for delay. Ann. Statist. 26 (1998), no. 6, 2179--2205. doi:10.1214/aos/1024691466.

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  • BASSEVILLE, M. and NIKIFOROV, I. 1993. Detection of Abrupt Changes: Theory and Applications. Prentice-Hall, Englewood Cliffs, NJ. Z.
  • BEIBEL, M. 1994. Bay es problems in change-point models for the Wiener process. In Change Z. Point Problems E. Carlstein, H.-G. Muller and D. Siegmund, eds. 1 6. IMS, Hay¨ ward, CA. Z.
  • BEIBEL, M. 1996. A note on Ritov's approach to the minimax property of the cusum procedure. Ann. Statist. 24 1804 1812. Z.
  • BEIBEL, M. and LERCHE, H. R. 1997. A new look at optimal stopping problems related to mathematical finance. Statist. Sinica 7 93 108. Z.
  • BOJDECKI, T. and HOSZA, J. 1984. On a generalized disorder problem. Stochastic Process. Appl. 18 349 359. Z.
  • BRODSKY, B. E. and DARKHOVSKY, B. S. 1992. Nonparametric Methods in Change-point Problems. Kluwer, Boston. Z.
  • CARLSTEIN, E., MULLER, H.-G. and SIEGMUND, D., eds. 1994. Change Point Problems. IMS, ¨ Hay ward, CA. Z.
  • CHOW, Y.S., ROBBINS, H. and SIEGMUND, D. 1971. Great Expectations: The Theory of Optimal Stopping. Houghton-Mifflin, Boston. Z.
  • CHUNG, K. L. 1968. A Course in Probability Theory. Academic Press, New York. Z.
  • DOOB, J. L. 1953. Stochastic Processes. Wiley, New York. Z.
  • DUBINS, L. E. and TEICHER, H. 1967. Optimal stopping when the future is discounted. Ann. Math. Statist. 38 601 605. Z.
  • DVORETSKY, A., KIEFER, J. and WOLFOWITZ, J. 1953. Sequential decision problems for processes with continuous time parameter. Testing hy potheses. Ann. Math. Statist. 24 254 264. Z.
  • JAMES, B., JAMES, K. L. and SIEGMUND, D. 1988. Conditional boundary crossing probabilities with applications to change-point problems. Ann. Probab. 16 825 839.
  • JANSON, S. 1986. Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift. Adv. in Appl. Probab. 18 865 879. Z.
  • KHAN, R. A. 1978. Wald's approximations to the average run length in cusum procedures. J. Statist. Plann. Inference 2 63 77. Z.
  • KENNEDY, D. P. 1976. Martingales related to cumulative sum tests and single-server queues. Stochastic Process. Appl. 4 261 269. Z.
  • KERESTECIOGLU, F. 1993. Change Detection and Input Design in Dy namical Sy stems. Research Studies Press, Taunton, UK. Z.
  • LEHOCZKY, J. P. 1977. Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probab. 5 601 607. Z.
  • LORDEN, G. 1971. Procedures for reacting to a change in distribution. Ann. Math. Statist. 42 1897 1908. Z.
  • MOUSTAKIDES, G. B. 1986. Optimal stopping times for detecting changes in distributions. Ann. Statist. 14 1379 1387. Z.
  • NEVEU, J. 1975. Discrete Parameter Martingales. North-Holland, Amsterdam. Z.
  • PAGE, E. S. 1954. Continuous inspection schemes. Biometrika 41 100 115. Z.
  • PELKOWITZ, L. 1987. The general discrete time disorder problem. Stochastics 20 89 110. Z.
  • REy NOLDS, M. R., JR. 1975. Approximations to the average run length in cumulative sum control charts. Technometrics 17 65 71. Z.
  • RITOV, Y. 1990. Decision theoretic optimality of the cusum procedure. Ann. Statist. 18 1464 1469. Z.
  • ROBBINS, H. E. and SAMUEL, E. 1966. An extension of Wald's Lemma. J. Appl. Probab. 3 272 273. Z.
  • SHIRy AYEV, A. N. 1963. On optimum methods in quickest detection problems. Theory Probab. Appl. 8 22 46. Z.
  • SHIRy AYEV, A. N. 1973. Statistical Sequential Analy sis. Amer. Math. Soc., Providence, RI. Z.
  • SHIRy AYEV, A. N. 1978. Optimal Stopping Rules. Springer, New York. Z.
  • SIEGMUND, D. 1985. Sequential Analy sis. Springer, New York. Z.
  • TAy LOR, H. M. 1975. A stopped Brownian motion formula. Ann. Probab. 3 234 246. Z.
  • WALD, A. 1947. Sequential Analy sis. Wiley, New York. Z.
  • YAKIR, B. 1996. Dy namic sampling policy for detecting a change in distribution, with a probability bound on false alarm. Ann. Statist. 24 2199 2214.