Minimax interpolation of random fields that are time homogeneous and isotropic on a sphere. (English. Ukrainian original) Zbl 0861.60062

Moklyachuk, M. P.

Theory Probab. Math. Stat. 50, 107-115 (1995); translation from Teor. Jmovirn. Mat. Stat. 50, 105-113 (1994).

Summary: The problem of the least mean-square linear estimation of the functional \[ A_T\xi= \int^T_0\int_{S_n} a(t,x)\xi(t,x)m_n(dx)dt \] of the unknown values of a random field \(\xi(t,x)\), \(t\in{\mathbf R}^1\), \(x\in S_n\), that is time homogeneous and isotropic on a sphere from observations of the field \(\xi(t,x)+\eta(t,x)\) for \(t\in{\mathbf R}^1\backslash[0,T]\), \(x\in S_n\), where \(\eta(t,x)\) is a random field that is time homogeneous and isotropic on a sphere \(S_n\) uncorrelated with \(\xi(t,x)\), is considered. Formulas are obtained for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional \(A_T\xi\). The least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional \(A_T\xi\) are determined for some classes of random fields.

MSC:

60G60	Random fields
60G35	Signal detection and filtering (aspects of stochastic processes)
62M20	Inference from stochastic processes and prediction
60G25	Prediction theory (aspects of stochastic processes)
93E10	Estimation and detection in stochastic control theory

Keywords:

least mean-square linear estimation; random field; optimal linear estimate; spectral characteristics; optimal estimates