This is an expository article. Here we show how the successfully used Kalman filter, popular with control engineers and other scientists, can be easily understood by statisticians if we use a Bayesian formulation and some well-known results in multivariate statistics. We also give a simple example illustrating the use of the Kalman filter for quality control work.

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In assessing the reliability of a system of components, it is usual to suppose the components to fail independently of each other. Often this is inappropriate because the common environment acting on all components induces correlation. For example, a harsh environment will encourage early failure of all components. A simple model that incorporates such dependencies is described, and several properties of this model investigated. Calculations are carried out for a parallel system of two components. Inequalities for multicomponent systems are suggested. The results generalize easily.

Library of Congress Cataloging-in-Publication Data Singpurwalla, Nozer D. Statistical methods in software engineering: reliability and risk 1N0zer D. Singpurwalla, Simon P. Wilson. p. cm.-(Springer series in statistics) Includes bibliographical references.

In this paper we show how several models used to describe the reliability of computer software can be comprehensively viewed by adopting a Bayesian point of view. We first provide an alternative motivation for a commonly used model, the Jelinski-Moranda model, using notions from shock models. We then show that some alternate models proposed in the literature can be derived by assigning specific prior distributions for the parameters of the above model. We also obtain other structural results such as stochastic inequalities and association, and discuss how these can be interpreted.

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