Moshe Pollak

Surveillance of a simple linear regression

This article considers an important aspect of the general sequential analysis problem where a process is in control up to some unknown point i = ν − 1, after which the distribution from which the observations are generated changes. An extensive sequential analytic literature assumes that the change in distribution is abrupt, for example, from N(0,1) to N(μ,1). There is also an extensive literature that deals with a gradual change in the case where the decision (whether or not a change has occurred) is based on a fixed set of observations, rather than an ongoing process of decision making every time a new observation is obtained. However, there is virtually no literature on the practical case of sequentially detecting a gradual change in distribution (visualize a machine deteriorating gradually). This article considers solutions to this problem. As a first approximation, the gradual change problem can be modeled as a change from a fixed distribution to a model of simple linear regression with respect to time (i.e., there is an abrupt change of slope, from a 0 to a nonzero slope). We study an extension of this case to a general context of sequential detection of a change in the slope of a simple linear regression. The residuals are assumed to be normally distributed. We consider both the case in which the baseline parameters are known and the case in which they are not. Finally, as an application, we monitor for an increase in the rate of global warming.

Nonanticipating estimation applied to sequential analysis and changepoint detection

Suppose a process yields independent observations whose distributions belong to a family parameterized by theta E Theta. When the process is in control, the observations are i.i.d. with a known parameter value theta(0). When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37-41] to construct a class of sequential tests and detection schemes whereby the unknown post-change parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixture-type rules are operationally or conceptually difficult to formulate. We exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a Gamma distribution, in both a univariate and a multivariate setting.

A simple comparision of mixture vs. nonanticipating estimation

When a problem entails an unknown nuisance parameter, estimating the parameter and (alternatively) integrating it out (with respect to some probability measure) are two of the approaches commonly used. Robbins and Siegmund (1973) proposed a non-anticipating estimation approach for problems which possess an intrinsic martingale structure which would be destroyed by the use of standard estimators. Here we compare this to the mixture (integration) approach in a sequential hypothesis testing context. We find that (in this context) the mixture is slightly better.