Maximum Likelihood Identification of Linear Discrete Stochastic Systems.

The method of maximum likelihood is applied to the identification of parameters in systems described by linear difference equations. The equations are assumed to be completely known except for the state variable coefficients, i.e., the state transition matrix, and, in certain situations, the initial conditions. The estimates are based on known normal operating input and on output measurements corrupted by additive Guassian noise. Maximum likelihood estimators of the parameters are developed for the following four cases initial condition known, initial condition unknown parameter, initial condition unknown random variable, and an equivalent equation-error model configuration. Finite sample and asymptotic properties of the estimators as well as computational aspects are investigated. The study is oriented toward real time applications. Application of maximum likelihood to the above four cases differs from the classical situation in statistics because the measurements are not identically distributed or are not independent or both. The resulting estimates are roots or cumbersome nonlinear equations.