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July, 1973 Estimation of the Covariance Function of a Homogeneous Process on the Sphere
Roch Roy
Ann. Statist. 1(4): 780-785 (July, 1973). DOI: 10.1214/aos/1176342475


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.


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Roch Roy. "Estimation of the Covariance Function of a Homogeneous Process on the Sphere." Ann. Statist. 1 (4) 780 - 785, July, 1973.


Published: July, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0263.62052
MathSciNet: MR334443
Digital Object Identifier: 10.1214/aos/1176342475

Primary: 62M15
Secondary: 60G10 , 60G15

Keywords: estimate of the covariance function , Homogeneous process on the sphere , spectral estimates , weak convergence

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 4 • July, 1973
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