An Estimate of The Parameters of A Complex Stationary Gaussian Markov Process

Parameter Estimation with Exact Distribution for Multidimensional Ornstein–Uhlenbeck Processes

“…Surprisingly, we have an exact distribution, not only an asymptotic property! This result was first formulated and applied in astronomy in Arato et al [5]. Complicated proofs can be found in Novikov [10], Liptser and Shiryayev [9], and Arato [1] [3].…”

Probabilistic analysis of the data on atmospheric angular momentum for January 1, 1980 to March 27, 2003

“…We do not assume this noise to be isotropic, i.e., gen erally, NOTE. In (Arato, Kolmogorov, and Sinai, 1962) and our previous works, the motion of the Earth's rotation axis under isotropic forcing is considered. In our present work, we abandon this assumption due to the pronounced anisotropy of the IERS data on atmo spheric excitation functions: the root mean square (rms) values of the functions and for 1982-2003 are 2.25 × 10 22 and 2.70 × 10 23 kg m 2 /s, respectively, i.e., their difference attains an order of magnitude.…”

“…In the probabilistic description of polar motion (Arato, Kolmogorov, and Sinai, 1962), the moment of forces causing this motion is a stationary random pro cess with a small correlation time, which is close to the white noise. Correspondingly, the Liouville equation, which renders the law of conservation of angular momentum, is a stochastic differential equation whose solution is a Markov process.…”

SEM Modeling with Singular Moment Matrices Part II: ML-Estimation of Sampled Stochastic Differential Equations

“…2 The model has numerous applications, for example, the statistical analysis of sunspots (Bartlett, 1946;Arato, Kolmogoroff, & Sinai, 1962;H. Singer, 1993), in physics (for an overview see Gitterman, 2005) or in psychology (coupled oscillators; Boker & Poponak, 2004).…”