On the Unimodality of the Exact Likelihood Function for Normal Ar(2) Series

Abstract

For stationary second-order autoregressive normal processes, the conjecture of uniqueness of the solution of the exact likelihood equations is examined. A sufficient condition for uniqueness is given; this condition is satisfied with very high probability if the number of observations is not extremely small. Moreover, it is shown that not more than two maxima may exist. Examples of data which actually produce a likelihood function with two local maxima are given. Keywords. Autoregressive model; first-order autoregression, maximum likelihood; second-order autoregression; uniqueness of the maximum likelihood estimate.

Exact Maximum Likelihood Estimation in Autoregressive Processes

“…Minozzo and Azzalini (1993) show that this condition is sufficient for guaranteeing the unimodality of the likelihood function for an AR(2) process. In general, we have not shown that this algorithm will always converge to the MLE.…”

“…That this stationary point is indeed the MLE is guaranteed by the condition D11(X)D,,(X) -Dl,(X)2 = (171.7918)(162.5333) -(82.3962)* > 0. Minozzo and Azzalini (1993) show that this condition is sufficient for guaranteeing the unimodality of the likelihood function for an AR(2) process. In general, we have not shown that this algorithm will always converge to the MLE.…”

“…Indeed, it may not, although we have found that for 'well-behaved' data, the algorithm often does converge quickly to the MLEs. Minozzo and Azzalini (1993) give an example of a data series ( X = [4.06 -5.27 5.24 -5.28 5.42 4.50IT) for which the likelihood function (assuming an AR(2) model) actually has two local maxima in the stationary region. When our algorithm is applied to this series, estimates of model parameters bounce about seemingly at random (both in and out of the stationary region) for many iterations, then slowly settle into a two-cycle, alternating between two solutions inside the stationary region, neither one corresponding to a true local maxima.…”

Modeling Covariance Parameters for Purely Autoregressive Correlated Longitudinal Data

“…In spite of the fact that there is no guarantee of a global convergence for these estimates, the results will be more encouraging when the covariance (correlation) structure of the data is well specified. Many authors have discussed the problem of multiple roots of the estimating equations, i.e., the stationarity of the parameter estimates ; (see Minozzo and Azzalini, 1993;Crowder, 1995;Desmond, 1997;Small et al, 2000;Crowder, 2001;Wang and Cary, 2003). Small et al (2000) considered Downloaded by [University of Florida] at 15:51 05 June 2016 Table 1 The simulated data of 20 subjects using (3.1) for 1 2 = 0 8 0 2 As an illustration, they considered different examples with multiple roots such as the estimation of the correlation coefficient, Cauchy location model, and weighted likelihood equations.…”

Modeling annual rainfall: a robust maximum likelihood approach

“…The estimating equation (4) has been derived in White (1961), Hasza (1980), and in Minozzo and Azzalini (1993). Martin and Yohai (1991) has suggested the use of the statistic medfX t =X tÀ1 g, where med denotes the empirical median, as a robust estimator of .…”