AbstractsMathematics

The problem of estimating the parameters of a forced discrete linear dynamic model is considered. The system model is conceptualized to include the value of the initial state as a parameter. The forces driving the system are partitioned into accessible and inaccessible inputs. Accessible inputs are those that are measured; inaccessible inputs are all others, including random disturbances. Maximum likelihood and mean upper likelihood estimators are derived. The mean upper likelihood estimator is a variant of the mean likelihood estimator and apparently has more favorable small sample properties than does the maximum likelihood estimator. A computational algorithm that does not require the inversion or storage of large matrices is developed. The estimators and the algorithm are derived for models having an arbitrary number of inputs and a single output. The extension to a two output system is illustrated; further extension to an arbitrary number of outputs follows trivially. The techniques were developed for the analysis of possibly unique realizations of a process. The assumption that the inaccessible input is a stationary process is necessary only over the period of observation. Freedom from the more general usual assumptions was made possible by treatment of the initial state as a parameter. The derived estimation technique should be particularly suitable for the analysis of observational data. Simulation studies are used to compare the estimators and assess their properties. The mean upper likelihood estimator has consistently smaller mean square error than does the maximum likelihood estimator. An example application is presented, representing a unique realization of a dynamic system. The problems associated with determination of concurrence of a hypothetical "system change" with a temporally identified event are examined, and associated problems of inference of causality based on observational data are discussed with respect to the example.