An extension of a change-point problem. (English) Zbl 1278.62020
Summary: We consider a specific classification problem in the context of change-point detection. We present generalized classical maximum likelihood tests for homogeneity of the observed sample in a simple form which avoids the complex direct estimation of unknown parameters. This paper proposes a martingale approach to transformation of test statistics. For sequential and retrospective testing problems, we propose the adapted Shiryayev-Roberts statistics in order to obtain simple tests with asymptotic power one. An important application of the developed methods is in the analysis of exposure’s measurements subject to limits of detection in occupational medicine.
MSC:
| 62C25 | Compound decision problems in statistical decision theory |
| 62L10 | Sequential statistical analysis |
Keywords:
Doob decomposition; change-point; classification; CUSUM statistics; likelihood ratio; limit of detection; martingale; martingale transforms; Shiryayev-Roberts statisticsSoftware:
NADAReferences:
| [1] | Page E. S., Biometrika 41 pp 100– (1954) · Zbl 0056.38002 · doi:10.1093/biomet/41.1-2.100 |
| [2] | Lai T. L., J. R. Statist. Soc. B 57 pp 613– (1995) |
| [3] | DOI: 10.1093/biomet/80.1.179 · Zbl 0771.62080 · doi:10.1093/biomet/80.1.179 |
| [4] | DOI: 10.1016/j.jspi.2004.11.010 · Zbl 1094.62100 · doi:10.1016/j.jspi.2004.11.010 |
| [5] | DOI: 10.1016/j.jspi.2004.01.006 · Zbl 1061.62028 · doi:10.1016/j.jspi.2004.01.006 |
| [6] | DOI: 10.1097/00001648-200211000-00021 · doi:10.1097/00001648-200211000-00021 |
| [7] | Helsel D., Nondetects and Data Analysis: Statistics for Censored Environmental Data · Zbl 1058.62111 |
| [8] | DOI: 10.1002/bimj.200610266 · doi:10.1002/bimj.200610266 |
| [9] | DOI: 10.1289/ehp.7199 · doi:10.1289/ehp.7199 |
| [10] | Hornung R. W., Appl. Occup. Environ. Hyg. 5 pp 46– (1990) · doi:10.1080/1047322X.1990.10389587 |
| [11] | Finkelstein M. M., Am. Ind. Hyg. Assoc. J. 62 pp 195– (2001) |
| [12] | DOI: 10.1093/aje/kwj039 · doi:10.1093/aje/kwj039 |
| [13] | DOI: 10.1021/ac00264a003 · doi:10.1021/ac00264a003 |
| [14] | Lai T. L., Statist. Sinica. 11 pp 303– (2001) |
| [15] | Robbins, H. and Siegmund, D. A class of stopping rules for testing parametric hypotheses. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 4. Univ. of Calif. Press. |
| [16] | Dragalin V. P., Econ. Qual. Control 12 pp 95– (1991) |
| [17] | DOI: 10.1214/009053605000000183 · Zbl 1077.62067 · doi:10.1214/009053605000000183 |
| [18] | DOI: 10.1016/j.jspi.2004.03.004 · Zbl 1061.62029 · doi:10.1016/j.jspi.2004.03.004 |
| [19] | DOI: 10.1214/aos/1176346587 · Zbl 0573.62074 · doi:10.1214/aos/1176346587 |
| [20] | DOI: 10.1214/aos/1176350373 · Zbl 0632.62080 · doi:10.1214/aos/1176350373 |
| [21] | DOI: 10.1214/aos/1176324467 · Zbl 0828.62072 · doi:10.1214/aos/1176324467 |
| [22] | DOI: 10.1214/aos/1176324712 · Zbl 0838.62065 · doi:10.1214/aos/1176324712 |
| [23] | DOI: 10.2307/2965584 · Zbl 0889.62070 · doi:10.2307/2965584 |
| [24] | DOI: 10.1198/016214503000233 · Zbl 1041.62069 · doi:10.1198/016214503000233 |
| [25] | DOI: 10.1214/aos/1024691095 · Zbl 0932.62090 · doi:10.1214/aos/1024691095 |
| [26] | DOI: 10.1214/aos/1176347761 · Zbl 0712.62073 · doi:10.1214/aos/1176347761 |
| [27] | Vexler, A., Liu, A. and Pollak, M. 2006. ”Transformation of changepoint detection methods into a Shiryayev–Roberts form”. The New York State University at Buffalo. Tech. Rep., Department of Biostatistics |
| [28] | DOI: 10.1214/aos/1176343001 · Zbl 0305.62014 · doi:10.1214/aos/1176343001 |
| [29] | DOI: 10.1093/biomet/67.1.79 · Zbl 0424.62015 · doi:10.1093/biomet/67.1.79 |
| [30] | DOI: 10.1016/0304-4149(94)90154-6 · Zbl 0789.62017 · doi:10.1016/0304-4149(94)90154-6 |
| [31] | Dragalin V. P., Econ. Qual. Control 11 pp 3-22– (1996) |
| [32] | DOI: 10.1214/aoms/1177696786 · Zbl 0239.62025 · doi:10.1214/aoms/1177696786 |
| [33] | DOI: 10.1016/j.spl.2004.10.035 · Zbl 1065.60039 · doi:10.1016/j.spl.2004.10.035 |
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.
