Nonlinear maximum likelihood estimation of autoregressive time series

Abstract

Abstract-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 highorder problems, we describe iterative algorithms for obtaining a ML solution.

Detection and estimation of abrupt changes contaminated by multiplicative Gaussian noise

“…Eq. (10) shows that the presence of a multiplicative white Gaussian noise (which causes a variance jump) leads to a non-Gaussian sufficient statistic. Note that d GH '0 ∀(i, j) which means that Z is a positivedefinite quadratic form in variables w(i).…”

“…Denote ln ¸(X"H G ) the Gaussian log-likelihood function for the vector X"[x(1), 2 ,x(N)]2 (with mean M G and covariance matrix G ) under hypothesis H G : where F"( f GH ) and G"(g GH ) are the N;N lower triangular matrices defined by [10] f GH "…”

Off-line detection and estimation of abrupt changes corrupted by multiplicative colored Gaussian noise

“…Under hypothesis Hi, the likelihood function for the where F and G are N x N lower triangular matrices defmed for instance in [8]. Under hypothesis H i , the inverse covariance matrix of the vector X can then be expressed as:…”

“…Ho rejected if Q (X) > S (PFA) (8) In (81, Q (X) = Qo (X) -Qi (X) where Qo (XI and Q1 (X) are the two positive definite quadratic forms…”

“…The Generalized Likelihood Ratio Detector (GLRD) estimates the unknown parameters under hypotheses HO and HI using the maximum likelihood procedure and uses these estimates in the Neyman-Pearson test defined in (8). The GLR test for our problem is:…”