Surface of Maximums of AR(2) Process Spectral Densities and its Application in Time Series Statistics

Alexander V. Ivanov, Nataliia M. Karpova


Background. In the problem on probabilities of large deviations of discrete time and sub-Gaussian AR(2) noise non­li­near regression model parameter least squares estimate a constant is determined that controls the rate of exponential convergence to zero of indicated probabilities.

Objective. The aim of the paper is to find the surface of maximums of AR(2) process spectral densities in the domain of its stationarity in explicit form.

Methods. The results were obtained on the use of methodology developed in the works by A. Sieders, K. Dzhaparidze (1987), A.V. Ivanov (1997, 2016) and standard Calculus methods.

Results. A complex formula that describes a continuous surface of maximums of AR(2) process spectral densities assig­ned on the stationary triangle of the time series of this type is obtained.

Conclusions. The obtained formula of surface of maximums of noise spectral densities gives an opportunity to realize for which values of AR(2) process characteristic polynomial coefficients it is possible to look for greater rate of convergence to zero of the probabilities of large deviations of the considered estimates.


Nonlinear regression model; Least squares estimate; Sub-Gaussian white noise; AR(2) process; Probabilities of large deviations; Surface of maximums of spectral densities


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