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We show how to define and then compute efficiently the marginal likelihood of an ARIMA model with missing observations. The computation is carried out by using the univariate version of the modified Kalman filter introduced by Ansley and Kohn (1985a), which allows a partially diffuse initial state vector. We also show how to predict and interpolate missing observations and obtain the mean squared error of the estimate.

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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. SUMMARYThe likelihood function for an autoregressive-moving average process is obtained by transforming the process to obtain a band covariance matrix whose Cholesky decomposition can be readily computed. Refinements are given for pure autoregressive and multiplicative seasonal processes. Evidence is presented to show that this approach allows more efficient computation than other methods proposed in the literature.Some key words: Autoregressive-moving average model; Exact likelihood function; Multiplicative seasonal model; Time series. ARMA(1, 0)

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