No abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARYRecently, methods for detecting unit roots in autoregressive and autoregressivemoving average time series have been proposed. The presence of a unit root indicates that the time series is not stationary but that differencing will reduce it to stationarity. The tests proposed to date require specification of the number of autoregressive and moving average coefficients in the model. In this paper we develop a test for unit roots which is based on an approximation of an autoregressive-moving average model by an autoregression. The test statistic is standard output from most regression programs and has a limit distribution whose percentiles have been tabulated. An example is provided.

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Let the time series {Y,: t E (1, 2, . . .)} satisfy Y, = pY,-, + Z , and Z, + Xf=, u,Z,-~ = el + XI"=, B,e,-,, where {e,} is a sequence of normal, independently distributed (NID(0, 0 ' ) ) random variables, and yo = 0. Associated with the Z, process are the characteristic equations inp + Eye, a,rnp--' = 0 and mq + Bjmq-J = 0, the roots of which are assumed to be less than one in absolute value. Thus, using the notation of Box and Jenkins (1976), we would say Y, is an ARIMA(p, 1, q ) process if p = 1. Under the assumption that p = 1, the limiting distributions of nonlinear least squares regression estimators of the parameters appearing in the preceding model are obtained. Regression t-type statistics for testing the hypothesis that p = 1 are discussed. Similar results are obtained for models that allow a nonzero mean. An illustrative example is given.

In the context of economic design, a two-stage model is proposed that utilizes continuously variable sampling intervals. Specifically, each successive sampling interval is determined by the extremity of the latest sample. Modeling the situation as a Markov chain, the hourly cost is developed for any arbitrary set of design parameters. This proposed approach is found to be more economically desirable, possessing a smaller average out-of-control production time, when compared with a standard two-stage approach where the sampling interval alternates between two fixed values.