Statistical problems for the Whittle field. (English. Ukrainian original) Zbl 0989.62050

Yadrenko, M. I.

Theory Probab. Math. Stat. 63, 179-183 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 163-167 (2000).

The Whittle random field \(\xi(x)\) on \(R^{n}\) is a solution of the stochastic partial differential equation \(\nabla^2\xi(x)-c^2\xi(x)=w'(x)\), where \(w'(x)\) is the “white noise” on \(R^{n}\). The author proves that if the Whittle random field \(\xi(x)\) is observed on the surface of a sphere with radius \(r\), then the linear predictor \(\hat\xi(0)\) of the value \(\xi(0)\) with the minimal mean square error has the form \[ \hat\xi(0)=[2r^{(n-2)/2}c^{(n-2)/2}K_{n/2-2}(cr)][(2\pi)^{n/2}L_{n}(cr)]^{-1}\int_{S_{n}}\xi(x)dm_{n}, \] where \(K_{\nu}(z)=\int_{0}^{+\infty}e^{-z ch(t)}ch(\nu t)dt\) is the modified Bessel function of the third kind; \[ L_{n}(z)=I_{(n-2)/2}(z)\{K_{n/2}(z)+K_{(n-4)/2}(z)\}-K_{(n-2)/2}\{I_{n/2}(z)+I_{(n-4)/2}(z)\}, \] \(I_{m}(z)\) is the modified Bessel function; and \(m_{n}(\cdot)\) is the Lebesgue measure on \(S_{n}\). If the random field \(\gamma(x)=a+\xi(x)\) is observed on the sphere with radius \(r\), where \(\xi(x)\) is the Whittle random field, then the variance of the linear unbiased estimate of the unknown mean value has the form \(D\hat a=\Gamma(n/2)L_{n}(cr)/(8\pi^{n/2}r^{n-3}c)\).

Reviewer: A.D.Borisenko (Kyïv)

MSC:

62M40	Random fields; image analysis
62M20	Inference from stochastic processes and prediction
60G60	Random fields

Keywords:

Whittle random field; linear prediction; mean square error; unbiased estimates