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The Gaussian log likelihood and stationary sequences. (English) Zbl 0879.62083

Subba Rao, T. (ed.), Developments in time series analysis. In honour of Maurice B. Priestley. London: Chapman & Hall. 69-79 (1993).
In a number of finite parameter problems in time series, a likelihood function is computed as if the process is Gaussian and this likelihood is maximized to obtain estimates of the unknown parameters. Our object is to get qualitative information about the global asymptotic behavior of this ‘Gaussian’ likelihood function. In particular we shall look at the sequence \((X_t)\) as an ARMA stationary \((p,q)\) process satisfying the relation \[ X_t-\varphi_1 X_{t-1}-\cdots- \varphi_pX_{t-p}= Z_t+\theta_1Z_{t-1}+\cdots+ \theta_qZ_{t-q}, \] where the \(Z_t\) terms are independent and identically distributed with mean 0 and variance \(\sigma^2\), with finite 4th moment, and the polynomials \(\varphi(\zeta)= 1-\varphi_1\zeta-\cdots- \varphi_p\zeta^p\) and \(\theta(\zeta)= 1+\theta_1\zeta+\cdots+ \theta_q\zeta^q\) have no roots on the unit circle.
For the entire collection see [Zbl 0846.00010].

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G10 Stationary stochastic processes