×

zbMATH — the first resource for mathematics

Nonparametric stopping rules for detecting small changes in location and scale families. (English) Zbl 1383.62156
Ferger, Dietmar (ed.) et al., From statistics to mathematical finance. Festschrift in honour of Winfried Stute. Cham: Springer (ISBN 978-3-319-50985-3/hbk; 978-3-319-50986-0/ebook). 251-271 (2017).
Summary: Nonparametric analogues of the Page-CUSUM procedure are constructed for sequential detection of location change in a distribution known to be symmetric about 0 and for detecting location change or scale change in an arbitrary unknown distribution. These stopping rules are defined on doubly-indexed stochastic processes whose weak limits are derived when there is no change and when there is a contiguous change. New fluctuation inequalities for rank sums are derived for proving tightness of these processes. In terms of these convergence properties, the nonparametric stopping rules are asymptotically equivalent to their parametric counterparts if the score functions used in both procedures are appropriate for the true density, but even otherwise, the nonparametric rules maintain their false alarm rates (due to the distribution-free property of ranks in the null case) and have good detection properties. The weak convergence results also show how the drift terms, which set in after a change occurs, and drive the underlying processes towards the decision boundary, slow down under model misspecification for both the parametric and the nonparametric procedures.
For the entire collection see [Zbl 1383.62010].
MSC:
62G99 Nonparametric inference
60G40 Stopping times; optimal stopping problems; gambling theory
62G20 Asymptotic properties of nonparametric inference
62L15 Optimal stopping in statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bhattacharya, P.K., Frierson, D., Jr. (1981) A nonparametric control chart for detecting small disorders. \( Ann. Statist.\) 9, 544-554. · Zbl 0503.62077
[2] Bhattachrya, P.K., Zhou, Hong (1994) A rank CUSUM procedure for detecting small changes in a symmetric distribution. \( Change-Point Problems, IMS Lecture Notes-Monograph Series\) 23, 57-65.
[3] Bhattachrya, P.K., Zhou, Hong (1996) A generalized CUSUM procedure for sequential detection of change-point in a parametric family when the initial distribution is unknown. \( Sequential. Anal.\) 15, 311-325. · Zbl 0876.62066
[4] Bhattachrya, P.K. (2005) A maximal inequality for nonnegative submartingales. \( Statist. Probab. Letters\) 72, 11-12. · Zbl 1065.60039
[5] Brown, B.M. (1971) Martingale central limit theorems. \( Ann. Math. Statist.\) 42, 59-66. · Zbl 0218.60048
[6] Chernoff, H., Savage, I.R. (1958) Asymptotic normality and efficiency of certain nonparametric test statistics. \( Ann. Math. Statist\) . 29, 972-994. · Zbl 0092.36501
[7] Doob, J.L. (1953) \( Stochastic Processes\) . Wiley.
[8] Hájek, J., Šidák, Z. (1967) \( Theory of Rank Tests\) . Academic Press.
[9] Hall, P., Heyde, C.C. (1980) \( Martingale Limit Theory and Applications\) . Academic Press. · Zbl 0462.60045
[10] Lorden, G. (1971) Procedures for reacting to a change in distributions. \( Ann. Math. Statist.\) 42, 1897-1908. · Zbl 0255.62067
[11] Moustakides, G.V. (1986) Optimal stopping times for detecting changes in distributions. \( Ann. Statist.\) 14, 1379-1387. · Zbl 0612.62116
[12] Page, E.S. (1954) Continuous inspection schemes. \( Biometrika\) 41, 100-115. · Zbl 0056.38002
[13] Wichura, M.J. (1969) Inequalities with applications to the weak convergence of random processes with multi-dimensional time parameters. · Zbl 0214.17701
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.