Nonlinear maximum likelihood estimation of autoregressive time series

In this paper, we describe an algorithm for finding the exact, nonlinear, maximum likelihood (ML) estimators for the parameters of an autoregressive time series. We demonstrate that the ML normal equations can be written as an interdependent set of cubic and quadratic equations in the AR polynomial coefficients. We present an algorithm that algebraically solves this set of nonlinear equations for low-order problems. For high-order problems, we describe iterative algorithms for obtaining a ML solution.