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

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 $$D\hat a=\Gamma(n/2)L_{n}(cr)/(8\pi^{n/2}r^{n-3}c)$$.

### MSC:

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