A homogeneous random process on the sphere $\{X(P): P \in S_2\}$ is a process whose mean is zero and whose covariance function depends only on the angular distance $\theta$ between the two points, i.e. $E\lbrack X(P)\rbrack \equiv 0$ and $E\lbrack X(P)X(Q)\rbrack = R(\theta)$. Given $T$ independent realizations of a Gaussian homogeneous process $X(P)$, we first derive the exact distribution of the spectral estimates introduced by Jones (1963 b). Further, an estimate $R^{(T)}(\theta)$ of the covariance function $R(\theta)$ is proposed. Exact expressions for its first- and second-order moments are derived and it is shown that the sequence of processes $\{T^{\frac{1}{2}}\lbrack R^{(T)}(\theta) - R(\theta)\rbrack\}^\infty_{T=1}$ converges weakly in $C\lbrack 0, \pi\rbrack$ to a given Gaussian process.
Publié le : 1973-07-14
Classification:
Homogeneous process on the sphere,
spectral estimates,
estimate of the covariance function,
weak convergence,
62M15,
60G10,
60G15
@article{1176342475,
author = {Roy, Roch},
title = {Estimation of the Covariance Function of a Homogeneous Process on the Sphere},
journal = {Ann. Statist.},
volume = {1},
number = {2},
year = {1973},
pages = { 780-785},
language = {en},
url = {http://dml.mathdoc.fr/item/1176342475}
}
Roy, Roch. Estimation of the Covariance Function of a Homogeneous Process on the Sphere. Ann. Statist., Tome 1 (1973) no. 2, pp. 780-785. http://gdmltest.u-ga.fr/item/1176342475/