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Multivariate Total Quality Control pp 1–86Cite as

Off-Line Statistical Process Control

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Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

Abstract

First part of this paper deals with tests on the stability of statistical models. The problem is formulated in terms of testing the null hypothesis H against the alternative hypothesis A. The null hypothesis H claims that the model remains the same during the whole observational period, usually it means that the parameters of the model do not change. The alternative hypothesis A claims that, at an unknown time point, the model changes, which means that some of the parameters of the model are subject to a change. In case we reject the null hypothesis H, i.e. when we decide that there is a change in the model, we concentrate on a number of questions that arise:

  • when has the model changed;

  • is there just one change or are there more changes;

  • what is the total number of changes etc.

The time moment when the model has changed is usually called change point. Aside testing for a change, our interest is to estimate change point(s) in different models. The least squares, M-, R- and MOSUM estimators are introduced and studied. Of course, we also estimate other parameters of the model(s), show approximations to the distributions of the change point estimators and show that the estimators of the change points are usually closely related to some of the test statistics treated in the first part.

Three types of confidence intervals are developed, one based on the limit distribution of the (point) estimators of m and two based on the bootstrap methods. All three methods are suitable for local changes while only the bootstrap constructions apply also to fixed changes.

The test statistics described below are typically certain functionals of partial sums of independent, identically distributed variables and their distribution is very complex. Therefore, we present selected limit results that form the basis for establishing the limit distribution of considered test statistics, functionals of partial sums and change point estimators.

Several Matlab codes, that implement selected methods described below and include detailed description and links to the previous sections, are presented to illustrate the possibilities of the studied methods. Complete Matlab codes are available from the authors on request.

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Authors and Affiliations

  1. Department of Statistics, Charles University of Prague, Sokolovská 83, CZ-18675, Praha 8, Czech Republic

    Jaromír Antoch & Marie Hušková

  2. Department of Mathematics, Czech Technical University, Thákurova 7, CZ-166 29, Praha 6, Czech Republic

    Daniela Jarušková

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  1. Dipartimento di Matematica e Statistica, Università degli Studi Napoli Frederico II, Via Cintia — Monte Sant’ Angelo, 80126, Napoli, Italy

    Carlo Lauro  & Vincenzo Esposito Vinzi  & 

  2. Faculty of Mathematics, Department of Statistics, Charles University, Sokolovská 83, 18675, Praha 8 - Karlin, Czech Republic

    Jaromir Antoch

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Antoch, J., Hušková, M., Jarušková, D. (2002). Off-Line Statistical Process Control. In: Lauro, C., Antoch, J., Vinzi, V.E., Saporta, G. (eds) Multivariate Total Quality Control. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-642-48710-1_1

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