A Note on Asymptotics of Estimators for Axially Symmetric Processes on the Sphere

Abstract Axially symmetric processes, those stationary in longitude but nonstationary across latitude, provide a flexible and physically meaningful class of models for global environmental data. Despite their wide use, the asymptotic properties of classical method-of-moments (MOM) estimators for these processes remain largely unexamined. In this work, we investigate MOM estimators of covariances and cross-variograms for axially symmetric Gaussian processes observed on regular latitude-longitude grids. First, we show that MOM covariance estimators are asymptotically biased. We then examine MOM estimators of cross-variograms, and prove that they are unbiased. However, using the block circulant structure of the covariance matrix and its Fourier diagonalization, we establish MOM cross-variogram estimators are not consistent. These findings illustrate intrinsic limitations of MOM-type estimators on compact manifolds and emphasize that Euclidean intuition does not carry over to spherical settings.