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.