The Annals of Statistics

Estimation of the Covariance Function of a Homogeneous Process on the Sphere

Roch Roy

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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.

Article information

Ann. Statist., Volume 1, Number 4 (1973), 780-785.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M15: Spectral analysis
Secondary: 60G10: Stationary processes 60G15: Gaussian processes

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


Roy, Roch. Estimation of the Covariance Function of a Homogeneous Process on the Sphere. Ann. Statist. 1 (1973), no. 4, 780--785. doi:10.1214/aos/1176342475.

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