Richard J. Meinhold’s research while affiliated with The Washington Institute for Near East Policy and other places

Kalman filter models based on the assumption of multivariate Gaussian distributions are known to be nonrobust. This means that when a large discrepancy arises between the prior distribution and the observed data, the posterior distribution becomes an unrealistic compromise between the two. In this article we discuss a rationale for how to robustify the Kalman filter. Specifically, we develop a model wherein the posterior distribution will revert to the prior when extreme outlying observations are encountered, and we point out that this can be achieved by assuming a multivariate distribution with Student-t marginals. To achieve fully robust results of the kind desired, it becomes necessary to forsake an exact distribution-theory approach and adopt an approximation method involving “poly-t” distributions. A recursive mechanism for implementing the multivariate-t—based Kalman filter is described, its properties are discussed, and the procedure is illustrated by an example.

In this expository paper, a new method for inference and extrapolations in certain dose–response, damage-assessment, and accelerated-life-testing studies is advocated. The method is based on the use of Kalman-filter smoothing. A distribution theory involving the double lognormal distribution is outlined, and some suggestions for implementing the method are made. An expository development of the backward recursive equations for Kalman-filter smoothing is given in an appendix.

The Kalman filter model, successfully used in a variety of situations, can also be used for inference from accelerated life tests. The use of this model calls for the specification of the “law of motion, ” and this means that a time transformation function such as the Arrhenius law, the power law, etc., must be specified. The ordered values of the stresses can be conceptualized as the ordered values of time, and thus inference about the life behavior at low stress can be viewed as inference about the “state of nature,” at a future time. The Kalman filter model has a fully Bayesian interpretation, and thus its use in accelerated life testing makes inference from such tests properly Bayesian and therefore coherent. This paper is preliminary, and we intend to present here the feasibility of such an approach.

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.

In this paper we present a commonly used model for describing software failures, and point out that some of the alternative models can be obtained by assigning specific prior distributions for the parameters of this model. The likelihood function of an unknown parameter of the model poses some interesting issues and problems, which can be meaningful addressed by adopting a Bayesian point of view. We present some real life data on software failures to illustrate the usefulness of the approach taken here.