1973/08/01 by Hirotugu Akaike · 1 citation
Engineering · Computer Science · Physics and Astronomy · Mathematics · #Control Systems and Identification #Gaussian Processes and Bayesian Inference #Scientific Research and Discoveries #Hessian matrix #Mathematics #Autoregressive model #Applied mathematics #Hessian equation #Likelihood function #Maximization #Autoregressive–moving-average model #Gaussian #Function (biology) #Expectation–maximization algorithm #Realization (probability) #Gaussian process #Estimation theory #Mathematical optimization #Statistics #Maximum likelihood #Mathematical analysis #Partial differential equation
paper · doi:10.2307/2334537
openalex publication_date 1973/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Closed form representations of the gradients and an approximation to the Hessian are given for an asymptotic approximation to the log likelihood function of a multidimensional autoregressive moving average Gaussian process. Their use for the numerical maximization of the likelihood function is discussed. It is shown that the procedure described by Hannan (1969) for the estimation of the parameters of one-dimensional autoregressive moving average processes is equivalent to a three-stage realization of one step of the Newton-Raphson procedure for the numerical maximization of the likelihood function, using the gradient and the approximate Hessian. This makes it straightforward to extend the procedure to the multidimensional case. The use of the block Toeplitz type characteristic of the approximate Hessian is pointed out.