The conditional, unconditional, or the exact maximum likelihood estimation and the least-squares estimation involve minimizing either the conditional or the unconditional residual sum of squares. The maximum likelihood estimation (MLE) approach and the nonlinear least squares (NLS) procedure involve an iterative search technique for obtaining global rather than local optimal estimates. Several authors have presented brief overviews of algorithms for solving NLS problems. Snezana S. Djordjevic (2019) presented a review of some unconstrained optimization methods based on the line search techniques. Mahaboob et al. (2017) proposed a different approach to estimate nonlinear regression models using numerical methods also based on the line search techniques. Mohammad, Waziri, and Santos (2019) have briefly reviewed methods for solving NLS problems, paying special attention to the structured quasi-Newton methods which are the family of the search line techniques. Ya-Xiang Yuan (2011) reviewed some recent results on numerical methods for nonlinear equations and NLS problems based on online searches and trust regions techniques, particularly on Levenberg-Marquardt type methods, quasi-Newton type methods, and trust regions algorithms. The purpose of this paper is to review some online searches and trust region's more well-known robust numerical optimization algorithms and the most used in practice for the estimation of time series models and other nonlinear regression models. The line searches algorithms considered are: Gradient algorithm, Steepest Descent (SD) algorithm, Newton-Raphson (NR) algorithm, Murray’s algorithm, Quasi-Newton (QN) algorithm, Gauss-Newton (GN) algorithm, Fletcher and Powell algorithm (FP), Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. While the only trust-region algorithm considered is the Levenberg-Marquardt (LM) algorithm. We also give some main advantages and disadvantages of these different algorithms.
This paper proposes a fast algorithm for the exact maximum likelihood estimation of parameters of multiple inputs transfer function models. This algorithm is a generalization of that proposed by Mélard (1984) which is a combination of an improved version of an algorithm of Pearlman (1980) which uses an algorithm of Morf, Sidhu et Kailath (1974) and consists to replace the (matrix) Riccati-type difference equation used in the Kalman filter by a (vector) Chandrasekhar-type difference equation with the quick recursion switching suggested by Gardner, Harvey and Phillips (1980) and an algorithm of Wilson (1979). Simulations and practical examples are used to illustrate the algorithm by comparing it with the method of Poskitt (1989), the generalized least squares method suggested by Sabiti (1993), and the nonlinear least-squares method of Box and Jenkins (1976).
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