×
Lorden, Gary; Pollak, Moshe

Nonanticipating estimation applied to sequential analysis and changepoint detection. (English) Zbl 1077.62067

Ann. Stat. 33, No. 3, 1422-1454 (2005).
Summary: Suppose a process yields independent observations whose distributions belong to a family parameterized by \(\theta\in \Theta\). When the process is in control, the observations are i.i.d. with a known parameter value \(\theta_0\). When the process is out of control the parameter changes. We apply an idea of H. Robbins and D. Siegmund [Proc. Sixth Berkeley Symp. Math. Stat. Probab. 4, 37–41 (1972)] to construct a class of sequential tests and detection schemes whereby the unkuown 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.

Cited in 23 Documents

MSC:

62L10 Sequential statistical analysis
62P30 Applications of statistics in engineering and industry; control charts
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62F05 Asymptotic properties of parametric tests
65C05 Monte Carlo methods

Keywords:

quality control; CUSUM; Shiryayev-Roberts; surveillance; statistical process control; power one tests; renewal theory; nonlinear renewal theory; Gamma distribution
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Dragalin, V. P. (1997). The sequential change point problem. Economic Quality Control 12 95–122. · Zbl 0914.62079
[2] Lai, T. L. and Siegmund, D. (1977). A nonlinear renewal theory with applications to sequential analysis. I. Ann. Statist. 5 946–954. JSTOR: · Zbl 0378.62069 · doi:10.1214/aos/1176343950
[3] Lorden, G. and Pollak, M. (1994). An alternative to mixtures for sequential testing and changepoint detection. Technical report. · Zbl 1077.62067
[4] Pollak, M. (1978). Optimality and almost optimality of mixture stopping rules. Ann. Statist . 6 910–916. JSTOR: · Zbl 0378.62071 · doi:10.1214/aos/1176344264
[5] Pollak, M. (1987). Average run lengths of an optimal method of detecting a change in distribution. Ann. Statist. 15 749–779. JSTOR: · Zbl 0632.62080 · doi:10.1214/aos/1176350373
[6] Pollak, M. and Siegmund, D. (1975). Approximations to the expected sample size of certain sequential tests. Ann. Statist. 3 1267–1282. · Zbl 0347.62063 · doi:10.1214/aos/1176343284
[7] 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. JSTOR: · Zbl 0732.62080 · doi:10.1214/aos/1176347990
[8] Pollak, M. and Yakir, B. (1999). A simple comparison of mixture vs. nonanticipating estimation. Sequential Anal. 18 157–164. · Zbl 1064.62556 · doi:10.1080/07474949908836428
[9] 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.
[10] Robbins, H. and Siegmund, D. (1974). The expected sample size of some tests of power one. Ann. Statist. 2 415–436. · Zbl 0318.62069 · doi:10.1214/aos/1176342704
[11] Roberts, S. W. (1966). A comparison of some control chart procedures. Technometrics 8 411–430.
[12] Shiryayev, A. N. (1963). On optimum methods in quickest detection problems. Theory Probab . Appl . 8 22–46. · Zbl 0213.43804 · doi:10.1137/1108002
[13] Siegmund, D. (1985). Sequential Analysis : Tests and Confidence Intervals. Springer, New York. · Zbl 0573.62071
[14] Siegmund, D. (1986). Boundary crossing probabilities and statistical applications. Ann. Statist . 14 361–404. JSTOR: · Zbl 0632.62077 · doi:10.1214/aos/1176349928
[15] 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. JSTOR: · Zbl 0821.62044 · doi:10.1214/aos/1176324466
[16] Stone, C. (1965). On moment generating functions and renewal theory. Ann. Math. Statist . 36 1298–1301. · Zbl 0135.18903 · doi:10.1214/aoms/1177700003
[17] Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia. · Zbl 0487.62062
[18] Yakir, B. (1995). A note on the run length to false alarm of a change-point detection policy. Ann . Statist . 23 272–281. JSTOR: · Zbl 0828.62072 · doi:10.1214/aos/1176324467
[19] 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. · Zbl 0937.60082 · doi:10.1214/aoap/1028903449
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.